I've caught a zillion oppossums in live traps. They seem to eat anything. I've probably caught more on peanut butter than anything else. But almost any table scraps from dinner will be fine.

Yeah, they hiss, but they get comfortable in the trap, you have to shake them out, and they scuttle off. I've never had one even look like it might be aggressive.

They do eat mice, which might be why you don't have any.

You do NOT want to poison it. If it dies where you can't reach it, the carcass will smell for a month, maybe more.

Hint: even if you are a very bad shot, it's pretty easy to hit one in a trap. I've killed them that way with low power pellet guns, though usually I release them.