Check your vehicle's owner manuals in regard go the charging of the batteries.
Some models come from the factory without the charge wire hooked up at the fuse panel (under the hood); mine didn't and the dealer did not hook up even though they said they did (and I didn't check) so a big surprise when I got to Glacier.
(Two days of thinking I was running the fridge but the fridge was draining the batteries.)
Also, on GM trucks, they advise to run with the lights on or in tow haul mode. (Daylight running lamps do not count.)
Reason is the computer looks at the vehicle's battery state of charge, ONLY, and if it is fully charged it will drop the charge rate to just keep the vehicle running. If the lights are on, the computer will keep the generator at a higher charge rate, thus supplying a higher charge rate to the trailer's battery(ies).
Tow haul does the same thing; however, tow haul is not recommended for a small trailer.