OK,
1) I know of no charger that charges at only 12.9 volts? So there's our first issue. Do you perhaps have the generator's 12v output charging the batteries directly? Another possibility is the generator is running , but you have an inverter/charger, and it is in "full charge" mode, and is turned off but monitoring the batts.
2) Your fridge does have 12v controls, so there is a small but contiuous draw there.
3) How many of these 6v batteries do you have hooked up together? 2, 4, 6?
4) That is the remote for the Inverter. Do you have the MM-AE Series inverter, with built in battery charger? Or does your converter have the charger option?
5) Using your hand held meter, you need to check the following voltages:
- output voltage of charger, whether its part of your inverter or converter
- overall bank voltage (main +, and main -, where power goes from batts to inverter)
- voltage across individual 12v strings (2-6v's in series)
- each individual 6v battery
--> V should be the same for overall bank, and each series string. V should be exactly half that for each 6v battery.
Yes, to be safe, put fridge in GAS mode when off grid. It runs on either gas, or 110v. 12v is continuous for the controls only. It is a two way fridge, not a three way. But if the inverter is on, and the fridge is plugged in to a 120v receptacle, it may switch to 120v, instead of gas, and draw down your batteries thru inverter power. So I would go with the gas setting, and keep an eye on your propane levels.