Fridge left on 110 means is runs on AC power when plugged in to shore power.
It probably swapped to 'gas' when unplugged from AC power.
Regardless.......both use 12 DC power for the control circuits.
Your truck can charge the trailer battery provided you have a 7 pin connector with a charge line AND your truck is set up to charge trailer.
Usually that requires a fuse/relay in trucks fuse box be inserted and some times a wire needs to be connected to that fuse position (Look in your trucks owners manual fuse section.......what is what and which one it is)
Also trucks charge system can really only slightly charge (basically maintain) trailer battery due to size of trucks alternator and the size/length of charge line.
Question.......
How do you know trailer battery was charged when you left dealer????
Because you actually tested voltage on battery
OR
because salesman showed you the 'battery monitor' and it was indicating full charge?? Those panel monitors will show full charge anytime you are plugged into AC power source
The only way to know is test battery voltage (no load/no converter charging) OR to have a good digital voltage monitor installed.
Get the truck 'tow' fuse checked
Get trailer battery recharged....either with a portable charger or via trailer converter if you can plug trailer into an AC power source.