The Havahart humane mouse traps are about $20, so that might get kind of expensive to buy more than one, but one should work. I have one and I’ve used it a couple of times, baited with a tny piece of bread smeaed with peanut butter. You are supposed to take the mouse far away from your house ( a mile or more) otherwise it will just come right back in unless you can find the entrance hole/holes and seal them.
There is one natural deterrent that I have heard of. I don’t know how well it works. Mice apparently don’t like the smell of peppermint. You take cotton balls and put a few drops of peppermiint essential oil (oil of peppermint) on them and scatter them around the house, or you can place the peppermint oil directly on floors, walls and other items, (although I might be afraid of damaging floors and such with oil). The scent should last a couple of weeks. Of course, you wouldn’t want your rabbit to ingest the cotton balls, if they were tempted by them, but maybe you could hide some under or behind the cage where your rabbit couldn’t get at them.