My ducts are all metal, so they can't chew but it still works as a pathway and a place for making nests. I plan to close all the registers permanently, since the furnace is no more - too much noise, 12V draw, and propane. Olympian 5 catalytic heater works fine, though my rig is a trailer, not 5-er, and shorter 26ft.
Plain old mouse trap is the only thing that works -
Trap. Don't put cheese, they "can" eat cheese but don't really like it, this is a myth. Put peeled nuts instead - walnuts, almonds. In kitchen and near heater registers. Make sure they are mice, not rats - the latter need a bigger trap
Rat Trap.
For long-term storage you might also put rat poison in small caps - mix it with rice, if you like an idea of having poison in your rig.
Neighborhood cat is keeping my rig clean now, but he is getting old. This is the best. He doesn't have to go inside, just walks around, sits under the rig, listening - mice are mobile, they go in and out and won't stay if there is a working cat nearby. Not all of them hunt - this is another myth.