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
2022 F-150 3.5 EcoBoost 4X4 Supercrew GCWR 19,500 157WB
Payload 2476 Maxtow 13,800 3.73 Equalizer 4 Pt Sway Hitch
2017 Jayco Jay Flight 24RBS
Old TV, 07 Toyota Tacoma, Double Cab, Factory Tow Pkg, retired towing at 229K. (Son now owns truck)