Virginia doesn't get cold enough to damage charged lead acid batteries, flooded or AGM.
Lithium it depends on if they have heaters, without heaters on lithium batteries they can be damaged by charging them if it's below freezing. With heaters check the manual of the batteries for how cold they can get and still charge. The lithiums should have a BMS to stop charging before damage but you can still discharge them in colder temps and discharge them to the point they shut down.
I leave my batteries in the trailer. I have a digital voltmeter in one of the lighter plugs in the trailer to monitor them. I plug the trailer into a an outlet every month or so for a day or two to let the batteries charge.
I think that the gold standard is to bring them inside after they are fully charged.
If the weather is not going to be sub-zero for extended periods of time, a fully-charged battery will be fine stored disconnected in the trailer. No need to bring inside, but they should be fully disconnected to prevent phantom loads draining the batteries.
If you have a multi-stage converter/charger you can just let it do its thing over the winter to keep your batteries topped off if you want to keep them connected. No need to buy/use another tender or battery maintainer, as a modern multi-stage converter/charger has this built in.
What not to do is just leave them in the trailer with the cables connected. Even if the disconnect switch is turn off, there are often still phantom loads connected which will eventually run the batteries to the damage zone.
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Richard
11018
1994 Excella 25 Follow the build on Gertie!
1999 Suburban LS 2500 w/7.4L V8
1974 GMC 4108a - Custom Coach Land Cruiser (Sold)
I live in Northern Jersey and never remove, or disconnect my AGM's and have not had an issue. I do, however, have solar on the roof and attach a suitcase panel to keep them topped.
If you actually have the unit plugged in, I am confident there is not need to disconnect them at all.
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"I thought growing old would take longer"
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