Their thought that the ground/neutral connection might be at fault makes a fair bit of sense. Most of our trailers are wired with a single, shared ground wire that runs from trailer to tow vehicle connection all the way around the perimeter of the trailer. Each individual 12v
light, fan, or other
electrical device then taps into this shared ground wire so that electricity that's "sent" to those devices on their own wire has a way back to the
battery. Add up all the different
lights, fans, pumps, etc. that tap into this single, shared wire and that's a lot of splices that can go wrong.
The proper way to troubleshoot a ground wire problem is by disconnecting the
battery and tow vehicle, then testing the ground wire connection at different points along the way to see where the problem(s) are, then fixing/replacing the specific wire splices or segments of wire that have gone awry. Your RV place may do the repair this way, or they may have an alternative "cheater's" solution.
The quick-and-dirty cheater's solution is to connect a new neutral wire from the TV connector to the tail end of the neutral line, giving the electricity two ways back to the
battery instead of one. This approach is a band-aid approach that will probably fix the lights, but in doing so it camouflages the actual cause of the problem and makes future
electrical system troubleshooting and repair more difficult.
From my description you probably have a clear idea which way I'd do this repair. Good luck.