Yeah I tohught he wanted to change up his style.
Overall while I get why you want a new weapon to be more student like, at the same time when your giving up an amazing item that many nobles would use if they had access also seems folly. I assume this change is so they can claim character "change" and some drama later where cause it's an inferior weapon it results in a death or major injury that wouldn't have happened with the other gun.
What BF28 does:
As mentioned by Tweight himself, BF28 doesn't have any particularly amazing abilities; it just increases the range of <Snipe> and makes it easier to aim. So it isn't a game-breaking World-class Item by any means. Its effects aren't even as great as the Ultimate Shooting Star Super, which increases its user's power and allows for its arrows to pierce through armour. (Overlord Reference)
Excerpt from Arc 2 Chapter 44 of the Web Novel:
(Tweight's monologue, explaining the functions of BF28)
Magical tools automatically activate magic without the user being aware of it, but this is not the case for magic tools with complex functions. BF28 requires the user to bear the burden of “forming the bullets” and “fixing your aim”.
Its only difference from a normal [Magical Wand] is that “the shots fly farther” and “it’s easier to aim”.
Reasons Tweight can't just use BF28:
(1) National Security
We know that matters concerning the Bureau (and the Organization) are kept classified for the sake of national security. Technology developed by the Organization, such as Miniaturization technology and the method to artificially induce an Eccentricity, is kept under wraps for the same reason. Since BF28 is a product of the Organization, it is also meant to be kept secret unless necessary (i.e., used for official missions).
(2) Possession of BF28
BF28 is technically owned by the Bureau, so Tweight can only use it when doing work for the Bureau.
Quote from Chapter 6 of the Manga:
Chris: Speaking of which, your beloved "Sniper Wand" is also under the Bureau's custody now.
In Chapter 10, the Bureau only gives BF28 to Tweight in order to kill Robert, since they "approved" of his operation.
Excerpt from Arc 1 Chapter 19 of the Web Novel:
(Chris giving BF28 to Tweight before the assassination of Robert)
"...Does this mean that the Bureau approves of this operation?" (Tweight)
"I can't give you a detailed explanation, but you can think of it that way. Otherwise, I wouldn't have brought that with me." (Chris)
Chris says as she points to the black bracelet I was wearing.
While for the Second Arc (Chapters 11 - 21 of the Manga), the Bureau only gives BF28 to Tweight for him to use for the duration of his escort mission, since it was officially handed to him by the Bureau itself. The reason he was able to continue using it even after he rebelled against the Bureau was because he was still holding onto it.
Quote from Chapter 11 of the Manga:
Chris: Also, since this mission is an official one from the Bureau, you have permission to use your beloved partner.
Since his school life isn't an official mission given to him by the Bureau, he isn't allowed to use BF28.
(3) Tweight's Wishes
In the same manner, Tweight avoids using combat techniques undeniably developed by the Organization. His reluctance to use these combat techniques, and other things developed by the Organization, is because it could lead to the revelation of the existence of the Organization and Tweight's relation to it, and as we know, Tweight wants to keep his past behind him (otherwise the premise of this story would just be thrown out the window).
It is for this reason that Tweight wishes to acquire a new method of fighting, one he can use in his everyday life as a student.
Excerpt from Arc 3 Chapter 7 of the Web Novel:
(Tweight's monologue when he decides to take concrete steps to learn new magic)
It’s been a while since the practice for the festival began, and my feelings of wanting to “try something new” still remain.
If I think about it, I’ve been spending my life as a student inconveniently. In order to hide my past as an agent of the Organization, I’ve tried not to stand out as much as possible, so when I fought in public, I had to use the minimum amount of force.
But things won’t stay this way forever.
If I spend my time in fear that my past will be exposed, I can’t call this my everyday life.
... (Omitted for Brevity)
“I want to learn new magic.”
When he does learn new magics befitting of a student, he employs it without fretting over whether or not it would reveal his past.
Excerpt from Arc 3 Chapter 16 of the Web Novel:
(Tweight announcing his intent to fight the intruders)
“This time, I will act as a student. I will not use any of the techniques I learned at the Organization, nor will I use BF28. And so, I will not seem suspicious.” (Tweight)
I inform Chris of my intentions.
“In the first place, the intruders are at the academy. If so, then it would be only natural for a student to notice and deal with them.” (Tweight)
“That’s…certainly true, but is it possible for you?” (Chris)
“Yeah. I’ve acquired the skills for exactly this kind of thing.” (Tweight)
I say while remembering what Sylphia-sensei taught me.
The reason he looked for a new wand is similar: because he wants to fight as a student.
Tangent (somewhat unrelated)
Unlike the previous arcs, where Tweight calls upon his past experience as an agent of the Organization to deal with the situations he finds himself in, this arc is one where Tweight actively wants to gain new experiences and make a clean break from his past, from being an ex-agent of the Organization.
This distinction between Tweight's identity as an ex-agent and his identity as a student is one of the main conflicts in this story.
Excerpt from Arc 2 Chapter 9 of the Web Novel:
(Tweight's comment on his identities as a student and an escort)
Student and Escort. These two positions have been switching at a dizzying pace.
This is why Tweight wanting to fight "as a student" is an important part of his development; it shows that he is finally taking more concrete steps to achieving his goal of living a normal, everyday life. Not one where his past "as an agent" continues to chain him down (as it had in the previous arcs).
After all, this story is supposed to be about Tweight trying his best to live a normal life, not him unnecessarily flexing his abilities at every minor inconvenience he encounters.
Counterpoint to having an admittedly inferior weapon, as a "Nerf":
There is a good point made that he is essentially nerfing himself, which could potentially lead to problems down the line. While, yes, this could very well be done to move the story along and bring about plot points that could easily have been avoided, I would like to point out that losing BF28 isn't a nerf for Tweight, as Tweight's strength isn't dependent on BF28.
To begin this segment, I would like to point out that for most of the 1st Arc, Tweight didn't even use BF28, yet still could handle himself in a fight, as seen in both the Entrance Exams and his Duel with Sieg.
In Tweight's duel against Sieg, where Sieg used Erdekaiser, a military-grade wand, renowned for its usage in the recently-ended Fourth Great Hero-Demon War. The reason I find this fitting is because we see that in this duel, Sieg is overly reliant on this amazing wand, so Tweight begins his counterattack, dealing with said wand by targeting a weakness in its design, leading to his "victory" (technically a draw).
To contrast this, I would like to bring up Tweight's fight against Oz, where before Tweight engaged Oz in frontal combat, Chris and Oz had made preparations to deal with Tweight sniping with BF28.
Excerpt from Arc 2 Chapter 49 of the Web Novel:
(Description of the Carriage Developed by the Bureau)
The carriage that 02 and the others were using now was custom-made by the Bureau. Not only were the horses performing well, the back of the carriage itself had a sturdy wall to prevent being sniped. During the Great War, agents of the Organization would often use these carriages.
However, there is a tiny gap in the wall for ventilation and visibility.
It was as the knight said, if the people riding in the back of the carriage were to get sniped, the shot would have to pass through this tiny gap.
Excerpt from Arc 2 Chapter 29 of the Web Novel:
(Countermeasures against 28's Sniping)
As they calculated backwards from the angle of 28’s snipe to find his location, the carriage stopped at a safe position.
While 28’s sniping is certainly miraculous, after a shot, there will be an interval before the next snipe.
The first shot is the most difficult to avoid, but the knight from the Kingdom of Arcadia dealt with it at the cost of his own life. With 02’s abilities, she was able to approach 28 within the span of that single interval.
As we can see, in both of these fights there were countermeasures for each weapon in question (Erdekaiser and BF28). While this brought about Sieg's defeat, in contrast, Tweight was able to hold his own against his opponent, almost bringing it to a draw despite his numerous disadvantages (poor condition, unfavourable terrain, bad match-up). Note that Tweight didn't even have BF28 when he fought with Oz, as Mize was holding onto it.
Sieg even puts it aptly in this very chapter:
If you merely wanted to win (the festival), you wouldn't even need a weapon, right?
So while having BF28 does contribute to Tweight's overall strength, it isn't like Tweight has no other options, and he is currently working on adding more abilities to his repertoire (such as new magic), which I believe makes him stronger (and more well-rounded) overall.
Minor Point (Why BF28 is only useful for Tweight):
It is mentioned that so far, Tweight is the only one who can use BF28 to its fullest potential. Even if nobles could somehow gain access to it, they wouldn't be able to use it properly, so it would be better for them to use a different Sniper Wand.
Quote from Chapter 6 of the Manga:
Tweight: ...Is someone inheriting that wand?
Chris: Not a soul. In fact, nobody out there can make full use of it other than you. In the first place, that weapon was created to be your own specialized weapon.
While Mize is shown to use BF28, she still isn't able to use it to its fullest abilities, which is why Tweight comments that a different Sniper Wand would suit her better.
Excerpt from Arc 2 Chapter 45 of the Web Novel:
(Mize having difficulties with some functions of BF28)
“Um, can’t I form the bullet first, before taking aim?” (Mize)
“That is impossible because of the way it works. Once you begin to form bullets, the focus of the [Far-sighted Crystal] is fixed.” (Tweight)
“But…I feel that my aim gets a bit off when I form the bullet.” (Mize)
“Yeah…That’s what makes BF28 difficult to use.” (Tweight)
This is the reason why the agents of the Royal Bureau of Counterintelligence weren’t able to take over this weapon.
“At the same time that the bullet is formed, the magic power needed to activate <Snipe> is also absorbed. If you unexpectedly resist because you can’t concentrate on forming the bullet, you will lose control over the wand.” (Tweight)
Ending Statement:
I'm surprised that I was able to make this longer than the previous one.
As always (this is just the third time, though), this entire write-up is merely based on my personal understanding of the story, so you may take it with a grain of salt, and I hope this explanation can help.
TL;DR:
Tweight needs to use a new wand, since in this Arc, he has resolved himself to fight as a student, instead of as an ex-agent of the Organization. Also because the Bureau took away his BF28 privileges. This is fine because Tweight isn't dependent on BF28.
As a side note, there is no mention of Tweight returning BF28 after the events of the Second Arc, and since he's technically still acting as Mize's escort while in school (which is technically a mission that the Bureau gave him in the first place), there is a slight possibility that he is still holding onto it. Also, his quote in Arc 3 Chapter 16 of the Web Novel, "...nor will I use BF28.", could imply that he has the capability to use BF28 (as in he has it on hand), but he simply chooses not to use it.