You can’t use Particle Collisions to do this since it’s not designed to work that way. The problem isn’t with collisions not working as such, since they work perfectly for their intended use.
In order to do this each individual particle would have to store tonnes of extra data, like it’s owner, instigator, damage etc. More often than not that’d overhead that nobody would ever want, making it pointless, memory consuming and hella-expensive. Additionally, any part of the simulation done on the GPU will not be able to update anything on the CPU. GPU particles are also bound to the GPU only, so they have no way to talk to the collision code running on the CPU.
Particle collisions are also expensive (CPU ones ((which yours are)) especially so), and they will get throttled if framerate drops too low, so it’s not reliable or scale-able either. Easy or not to do so in Unity, that’s a horrendously expensive way to deal damage to something regardless of whether the engine let’s you do it or not.
A better approach would be to create a simple sphere or capsule primitive (or a few if you must), that roughly covers the area of the particle system. Another way would also be to spawn several instanced ‘projectiles’ within that volume that are bound roughly to it’s shape. You’ll have to do some tweaking to get that to feel right, but that’s really the only way to do it.