I might make 2 animators, one one the gun and one on the character
if necessary, the gun is a parent of the hand, or for animations temporarily unparented, since for generic animation the transform hierarchy has to match the one in blender too
and I havea script that controls that the same animations on both animators are played simultaneously, or paremters set etc
I mean you already said you have 2 animations, i don't really know how to handle it in blender.
alternatively you can make just one animation or rig for everything, in unity, if you use humanoid animation type, you go to the import settings and there should be an avatar mask and you enable additional bones there, and also in the character you animate, it should also have these bones. if not, that's tricky. one time i imported a humanoid in blender, added the transform hierarchies i needed, and exported again, so that in unity i could set an avatar mask that included these bones. because avatar masks go by the source transforms of the imported object as references, you can't really add bones at runtime afaik
if you look up avatar masks and import settings