"Y-you... What the fuck are you?!"
"A blue control mage."
Man holding those counterspells to clear a path for his RG burn "Cast Fist" spell.
In the meantime, let me dig up and elaborate why not killing Barbaros is in-line with Zagan's character (and his inner-demon for those savvy in storywriting). But tl;dr, as someone mentioned above, Zagan is simply a loner who really wanted a friend to hang around, according to LN. What I will say below is from my perspective as a casual manga reader who don't read the LN (or the anime, if any is airing as of time of writing).
1. Zagan is a closed shut-in, as beautifully said by Nephie (ch. 12 pg. 9). His old life of being a street runaway who got used to stealing and getting beat effectively close the door to any normal form of connections. He has thrown away the way of living normally and connecting to people, and delved into the way of the outlaw, living off of others just to survive, at the cost of causing harm to the other party.
2. And then, Andras the Resentment happened. Sacrifice is seemingly one way to acquire power by research of magic majutsu (in other words, a lab rat), so this is a one-way road to a certain demise for Zagan. At this point, there is no hope left for him. He is just a street brat, now squirming in-between the fingers of a mage. He is experiencing his first near-death experience in such a young age, and it is only by the impulse of youth Zagan managed to kill Andras, and thus, survived. This gave birth to his principle, his ideal, that "to survive, go stronger". And with it carries the implication that you cannot rely on others. Again, Zagan went knee-deep... no, belly-deep of being a shut-in. But then, what happened when
3. Zagan saved a caravan? (ch. 7 pg. 21); why would someone who "won't rely on others" to "get stronger and survive" save "others" by himself, a "weak" existence that wouldn't "survive" otherwise? It is precisely because point 1) and 2). They are "weak", just like him before. They wouldn't normally "survive", just like himself when captured by Andras. And to no one's surprise, the girl objected to Zagan's ideals "Why is it important to get strong? Can't the weak go on with life?"; this strikes Zagan and even solidifies his character as
4. pragmatic and cold, traits born out of need to survive. He dislike being weak, because it gets him dead, yet when he sees others in exactly the same position, he can't let his cold side to the surface; because he sees himself in their position. He projects himself into the position of those in distress. And when he is in distress, he wants to get out of it.
5. Thus, Zagan helped others. He helped others as he would see himself being saved by others. This is his inner demon: weak and alone, without anyone else to lean your head on. He helped others so his self-projection can "survive". You may see this as a selfish way to fulfill his desire, as naive way to fulfill his inner wishes, YMMV. However, this inner-demon and principles that are in conflict with one another gives birth to the character that is Zagan: someone who lost an important aspect of himself for the sake of survival.
So why would Zagan dismiss Barbaros who outright said "I will kill you and won't stop until I achieve so!", implying he will do everything he can to make Zagan suffer, including doing nasty things to Nephy? Why is Zagan not threatened which such threat, knowing he is the weakest of Demon Lord and its implication of being in the Church's crosshairs (and prolly other Mages')? Because it is in-line with his inner-demon: weak and alone. If he kills Barbaros right there, he fails to solve his inner-demon of being alone. In a way, it is also a show of weakness, being unable to handle a threat as they arise, so you should weed out everyone else while they're weaker than you. But most importantly, he simply wants to stay friend with Barbaros, the only person he made a connection until Nephy. He simply wants to fight against his inner-demon.
Call him naive, call him childish, call him idiot, because he is. It is dangerous to do so, and perhaps killing Barbaros is the correct course of action, yes. But remember, Zagan is a character, and he has grown out of his shell. He found Nephy, someone that fills an important aspect of his character that were lost once, and he don't want to lose it again by killing Barbaros, the only person who he had a toast with before.