I can't tell, but I suspect that if there is a converter, there is a connection in the converter between 12VDC white return wiring and the 120VAC green ground, with the green ground being connected to the trailer frame. On
Scamp, that connection is in the driver's side wheel well.
If the running lights are getting their return via a white wire to each outlet and then back to the TV connector, that is a better way than relying on frame ground/return, but a break in that wire may still result in strange
lighting symptoms, regardless of whether it gets to the TV negative terminal by frame ground or direct connection.
Scamp's wiring system uses a white wire return. Regardless, hooking up a long wire from TV
battery to
light fixture should yield info as to whether this connection is a problem.
If the tow vehicle has turn signals on one set of bulbs (usually yellow) and
brakes on another (usually red), and the trailer is using one filament on the double-filament bulb (typically 1157; with other filament for running lights), then a diode block or relay system is needed on the tow vehicle to preclude current from the brake or turn signals from back-feeding from the trailer bulb to the other truck bulb(s). Stepping on the TV
brakes may cause the TV
brakes AND both yellow bulbs to
light.
Late model vehicles with tow packages, or after-market harnesses specifically made for the TV, have these provisions in them. I think Toyota is an exception, with the Toyota-made harness a necessity.