We had a Shar-Pei who loved to catch mice (it is one of the jobs they were originally bred for). She never seemed to have a problem related to killing and eating mice, and in fact had a lovely time tossing them in the air for a bit before delivering the coup de grace.
We also have what we describe as a “dachs-weiler” (looks like a rottwieler but only has about 5 inches of ground clearance) who routinely catches mice (and the odd rat) in our laundry room. He doesn’t eat them but leaves them for me to step on in the morning.
It sounds as if your dogs would eventually catch the mouse in your garage, but be aware that dogs don’t care about blood smears or shifting things in order to get to their target. Because they are up-to-date on shots there may not be a health problem depending on where you live, but in the interests of time, efficiency and neatness, you should consider a baited trap of some sort. There are several kinds these days, none are expensive, and many can be found on the shelves of the typical grocery store.
If you really want the dogs to do the work for you, I suggest you ask your vet what the potential risks would be in your area, and then be prepared to clean the garage.