Hanta - is that really a health problem in your area? That sounds somewhat paranoid to me.
Here it's rats that might cause problems if left unchecked, nothing serious nowadays, something different in 3rd world countries and in history.
Where there are rabbits, or any lifestock, outside, there always are mice. There's always some food to pilfer, and no matter if you notice it or not, mice are there. With not any problem I've heard of. Other than rabbits wont eat soiled hay. I'm so happy about the snakes in my barn, but they wouldn't be there if there weren't mice.
So I woudn't worry much about second generation mice in a different cage that are not soiling the rabbits or your food. Maybe one thing to read up about would be mites.
Apart from that, I had some "wild" mice indoors, House Mouse, Mus Musculus - awful, they do stink to heaven, I can tell if there are any when I enter a barn, garage or stable, very distinctive smell.
Wood mouse, Apodemus sylvaticus, much better in that regard, also field mouse, Microtus arvalis.
It may be different in other climates, where everything drops dead when encountering another spieces, but here that's pretty much a nonissue. If you really would need to wear facemasks and follow biohazard protocols to stay safe I would pile up a pyre and get rid of it.
Anyway, there sure are good forums about mice out there, where you can find solid advice.