Some notes come to mind...
1. All modern TV's that I've seen that have a trailer towing harness' equipped from the factory, use a relay (and fuse) in the POS trailer charging lead that ONLY allows a connection when the TV's ignition switch is ON...when the TV's engine is OFF, it's impossible to discharge the TV battery through this lead!
2. It's possible that if you have an aftermarket or repair shop installed towing harness - this RELAY/Ignition could have not been installed, IE they hooked the trailer charging lead directly to the battery POS...NOT a good idea!
3. When your 7-way trailer cord is hooked to the TV, the trailer brake controller is now hooked into the circuit, and any irregularities in that device could cause discharging of the TV's battery...Also, if there is any feed through to the brake circuit in the trailer itself, via a brake controller defect or adjustment, you may indeed discharge the TV battery...
4. You mentioned the refrigerator operation....if you have a 3-way refer and you're trying to run it powered by
12 volt DC, you'll run down your batteries in short order! the 12 VDC operation is only for operation when running down the road with the TV's alternator providing the necessary current to run the high current heating element in the refer...
5. If you only have a 2-way refer (propane/110VAC), and it won't get 'cold' when plugged into AC shore power or running on Propane - you've probably got problems with the refer's cooling unit, and will need replacement or an exchange cooling unit installed...
6. You mentioned hooking the trailer to the RV 'cooked' the TV battery??? Again, you've got some big TV tow harness wiring problems! I can't imagine any way the TV battery would become 'hot' by hooking up the 7-way trailer cord...
a. IF you have a non-relay connection in the TV charging lead (as explained above), AND you're hooked up to AC shore power, the converter/charger will indeed be charging both the trailer and TV batteries - over a long period of time this 'could' overcharge all the batteries...
b. I'd suspect the brake controller before the charging lead, which could discharge the TV battery, but not necessarily cause the battery to be 'HOT', unless there was a really LARGE amperage drain...having the brakes full-on would mean a really high amp load on the TV battery!
I leave my trailer cord hooked up for days, sometimes, and never have had the TV battery become discharged...check out all these circuits, one at a time till you find the problem...
Ray