Houdini the mystical rat!

[JLC]

Can you poison the peanut butter he keeps licking off the snap traps?

Have you tried a have-a-heart trap? It's a live-catch trap you can get at the hardware store. Worked well for me when I had live rats escape from an unhungry ball python. (I wasn't all that good at catching them to remove them from the shallow tub and they'd jump out. )

[Rhasputin]

Havahart traps work great too. I use the mouse sized ones all the time.
They're a fantastic investment. I don't even bait mine. Just put them up against the wall, and the mice eventually run through them.