urban catch and release

I have a little house mouse problem, there's one in the tank (and he's really cute). I also have a cat problem - Levi keeps catching them and then all he will do is play with them and let them go in the house again. The last couple of nights he's caught one and I put it outside, knowing it would just come back in.

I don't have it in me to kill the little creatures when I take them away from the cat, my girlfriend suggested caging them and releasing them away from my home. From what I've read online I need to go at least two miles away, and release them near (abandoned) buildings rather than in the woods, as city mice don't fair well in the country. I don't think this one is a nursing mother - I didn't see any exposed nipples, so hopefully she's not leaving a nest of helpless babies behind. If anyone has any suggestions I'd love it.