How do you know the truck does not charge the 12v trailer battery? Get that fixed while at the dealer.
Detectors and some electronics will run your battery down when not connected to a power source. This is normal and common. You need to disconnect a battery terminal or install a switch to prevent this from happening. Typical for a dealer to let the battery go dead and sell you a new battery sooner than later. Overly discharging a battery as you describe will permanently reduce capacity and shorten the battery life.
Dealer may use a jump box direct on the battery or just a common shop battery wheeled around with jump cables just to temporarily operate the 12v systems for the fix or get you off the lot.
The 12 Volt Side of Life