Airsoft is a big no no. That's what we use to get the red squirrels from trying to burrow into our roof. (One actually managed to eat through the roof and ended up in our kitchen. So we needed to do it)
When my cat decided to be an asshole and bring mice/small birds into the house half-alive, I brought it outside and used a 495fps 0.177, and ballistic ammo (the expensive stuff at Canadian tire) with a point blank shot through the bottom of the jaw into the head, or a shovel. Whatever was easy to get at so it didn't need to suffer.
For larger stuff (my dog loved to bring moles in), I used a spade and a very hard hit on concrete to put it out of misery. Nothing that'd risk putting it in more pain.