When your TT is hooked to the tow vehicle (TV) the TT draws off both batteries if you do not have an isolator in the rv or TV .
They are normally installed in the TV , some have them from the factory not sure about Tundra.
Once you park and shut off the truck and deploy slide , I would unplug the TT from the truck if you plan to stay parked for awhile .
An isolator is a gate that keeps the TT isolated from the TV and will not run down the TV battery. Just remember to plug it back in before you travel again.
Isolator solenoids are very inexpensive and can be installed on the trucks firewall. It offers protection in case you have a weak RV battery, or a cell goes bad due to a floating short, causing a draw on the TV battery.