Had issues with squirrels setting up camp on top of our warm inverter a couple of years back. Parked for years, I'd leave the system plugged in over the winters when solar was scarce to keep the batteries up. In summer solar alone was enough, so the inverter didn't get warm as I wasn't plugged in.
S*** (and p***) happens, pun intended. Cost me time and effort to get the inverter engineered out of the coach, cleaned, and then money to get it checked out, and reinstall it. Then the critters got in again, rinse and repeat, except the empathetic repair company didn't charge the second time. By then Magnum parts were not to be found anyway, so I lucked out that they could get it going. Getting the thing off the bay ceiling and back, twice, was a trip for an old codger!
The only place I could fathom the critters were getting in was a wiring loom port along the curbside rail into the adjoining waste tank and utilities bay. I tried my best to spray anti-rodent expanding foam up in the wire port, but couldn't really get at or see it very well, and mostly made a sticky mess. But I thought I had things licked... until this winter when I found a couple new turds in the inverter's bay, and the inverter quit again. Welcome to RV ownership! I've yet to summon the wherewithal to empty the storage bay and crawl in there again. We'll see. I've been checking and keeping the batteries up with a new nifty charger I researched and like, a Noco Genius10. With battery Mains off, I only need to hook the charger up once every 4-6 weeks just as a check and top off (or chassis pair equalize), but while switched off the batteries stay up pretty much on their own.
As to your "Battery Cut-off" switch, aka Salesman's switch or labeled "Coach Power", it is a common trouble spot. Experienced advisors will tell you to leave it on; if you want to stop 12v drainage, shut off the Main switches in the battery bay instead. Tape the entry door Salesman's switch down, cover it so grandkids can't mess with it, or jumper its connections so it can't be used on purpose or inadvertently. Ken Carpenter, revered Beaver Coach service advisor much missed, once warned me in no uncertain terms (but with a chuckle) that if I ever shut that switch off he'd know it and come find me! The switch, that feeds through a Ford-style solenoid on the battery/fuse bay sidewall, is apparently susceptible to wear and leaving folks without certain interior amenities. If it wasn't the switch wearing out, it could be the solenoid, which was part of our issue at the time Ken chimed in. Owners would be in the habit of hitting the switch when leaving the coach for dinner out or longer periods, obstensibley to save battery strength. But in time the switch or solenoid failed to actually shut off, leaving batteries connected to interior features that slowly drained things. I'm no expert here, but would expect to find voltage at the small + terminal of that solenoid if the entry door switch was on, so you can tell which position is which. Others here may better advise.
Joel