I tried to figure out some bugs about mobs, because I love mojang so I want to help them, and I trapped mobs and let anvils fall on them, and noticed, that bats don't have any kind of affect when anvils fall through them.
Causes: Bats are actually are flying, (same problem, if you have fly mode, and you are in survival, anvils are don't have any affects on you), and the anvil is ONLY have damage if it hits the ground.
Solution: You can't solve the problem by hand, but mojang can make the anvils so they have damage in air.

Thank you