Isn't the OP interested in having the rig plugged in and then having the ability to go out periodically operate moderate loads without worrying about running the batteries down?
"Home" could be Fargo North Dakota or Stovepipe Wells, Nevada. It's kinda tougher to help someone when a person is forced to guess things. Where it's really cold is another universe away from the subtropical southern climate.
A 12/3 extension cord can power a converter, plus heater and lights plus radio.
I would leave the cord plugged in the whole time I was in the rig, plus leave it connected overnight, if it has to be removed.
Unless it's a really ancient trailer with a Fred Flintstone converter, the converter should handle occasional use just fine. Problems can arise if a person is careless and leaves loads switched in then has a break in the power cord for some reason. This will ruin the batteries, utterly, especially if it freezes.
Hah! I remember buying those 100 feet orange 12/3 extension cords in Costco for $39.95 How times change!