As mentioned you will get a lot of opinions regarding the storage or the TT and batteries. If you as 20 people their thoughts you will get at least 20 different opinions regarding battery storage. Which one is the correct answer, that depends on a lot of things.
Some members take their batteries out at the end of the season and put them on a work bench in the garage with a 3 stage smart charger.
Some members take them out because they do not feel safe leaving their battery investment in a storage area, in fear that someone will take them for their own use.
Some members can not take them out because they suffer from lack of youth (yours truly), or have injuries, so they just charge them up to 100% and pull the 30 amp fuse between the battery(s) and the TT.
A member took his batteries and put them in the storage area, not a recommended way to store them.
Some members are lucky and have a Shore-Power hookup at their storage.
Other members purchase a smaller SOLAR panel and SOLAR charge controller to keep the battery topped off over the winter in storage.
And the list goes on.
Leaving a FULLY Charged battery, full on water and with the 30 amp fuse pulled over winter, there should be no issue, as the discharge process slows down to a snails pace in cold temperatures, but if it is an older battery I would pull it and take it back to the house and put it on a SMART charger.
I LOVE my SOLAR, for the last 5 years it has kept my Trojan T145 (260 Ah) batteries in great shape. The TT's battery charge controller has not been turned on since I added SOLAR.
Tip: After each outing, if you do not have shore power while not using the TT, PULL the 30 Amp fuse between the battery and TT. It only takes a week or 2 to totally drain your battery if you do not. There are parasitic loads that constantly put a drain on the battery (CO detector, radio circuit board, fridge circuit board..)
The decision is up to you,
Don
My Registry
RVing with SOLAR