I has the same thing happen to me a about 5 yrs ago. I ordered white duct tape online from Graingers. As mentioned, I have had to replace it once because of drying out and yes it does make a sticky mess. I just cut the same size patch (on both sides) to go over the old patch area which worked out well. I keep the roll of duct tape in the camper. overall the patch holds out well.
I would bet a cup of coffee the mice are coming in from under the floor deck, either from an uncalked utility entrance or an unfinished opening of the floor. I had both, caulked everything and mice never returned.
Mike