Last year I brought my TT home in December and I sat in the side yard until late February. The TT was unplugged and I put the battery in the mud room under the sink.
This year I parked the TT in the backyard where I have power and kept it plugged in. I left the battery on the TT and check it monthly for water.
My TT has a variable charger as part of the 12 volt converter which reduces to a trickle charge when the battery is fully charged.
There are two pit fall to watch out for, first I know from experimentation that the phantom loads in my TT will draw down the battery if a few days if I lose shore power. Second not all converters will trickle charge, some will maintain enough voltage to boil the battery dry.
I do not know if there is an advantage to keeping the TT plugged in all winter but I do not think it hurts to keep the TV, microwave and other phantom loads warm and toasty.