They aren't exactly harmless in that they do cause electrical fires by chewing the insulation off wiring.
I'd buy a bunch of old-fashioned snap-whack-bang traps. They are cheap and they work. Put them in places the cat can't reach. Mice like peanut butter a great deal.
Good use for the recalled peanut butter
I use the chunky type. I wedge a peanut into the trap and then smear the peanut butter on. Gets them every time.