@themanj well, to answer your question. Understand what was the principle behind weapons: to deliver a concentrated force enough to cause damage, potentially disabling, crippling, or killing whatever was struck. Now, with force, there's an equation: F=MA (Mass + Acceleration). Modern weapons operate on a principle of high acceleration projectiles with little mass to deliver devastating amounts of force.
Now, the big problem is that you'd need a good stable platform to be able to fire those anti-armor rounds from. Coupled with the major possibility that these things are
highly mobile, anything short of an exoskeleton to offset the weight and still grant some maneuverability won't work.
Now as for the round makeup...
The most likely candidate for making powerful rounds against them would be something fired from a rail or gauss. Unfortunately no matter how much miniaturization you manage to accomplish, those things aren't projected to reach fallout-verse sizes for a while, especially coupled with how big the energy requirement is needed by the system to accelerate the projectile to the necessary speeds. This also includes the problem of rate of fire; since you need to run the magnetic force all throughout the rail, there needs to be a time for it to return to its magnetically neutral state for the next projectile, hampering rof. So, it's going to be heavy and slow.
Using ballistics, you'd have to tweak alot the formula of F=MA. Three possibilities do arise: increase the mass of the bullet, increase the size of the bullet, or increase the initial acceleration via the propellant.
Increasing the mass of the bullet has already been underway, with the use of depleted uranium rounds. The problem there is figuring out how much mass is needed to be able to get a force capable of penetrating those demons. If it can be figured out, you can create a bullet that is compact, heavy enough, with enough kick to send those demons back where they came from. Unfortunately this would take several years of study to figure it out and it could be too late when it's figured out. Not to mention the logistics of getting the material with enough mass to do the job.
Increasing the size tries to take the mass equation in another direction with what materials available. However, the answer to getting enough oomph might be too big that it'll end up being like trying to shoot a fly with a bb gun: doable, but hard. ...Especially when said flies are ravenous swarms.
Advances in propellant tech might be the answer. However the rounds maybe unstable, and might not actually withstand the forces created by propellants.
It'll require a mix of a 3 of what I said, but it's most likely by the time the exact mix is found, majority of the world would've been overrun and the answer might be too unwieldy and bulky to use.
If that's the case, using soldiers in exoskeletons with really pointy sticks is the way to go, until you find a way to augment soldiers...which is what they did with the Yuusha Project.
That's the most likely reason for their actions...though they skipped a couple of steps that I mentioned.