First let me say that grounding the trailers electrical system is an old and poor way of doing so. Running actual ground wires is a better and safer way to do it. Think of losing the ground contact through the ball (rusty) while driving at night. Guess what happens to all your trailer lights? Yep, off!!!
Okay on average an actual trailer ground wire is 14 gauge (could be 12 or 16). I'm sure the safety cable/chain is much bigger than that. If not there's no safety involved LOL. There's no way the cable can be getting hot because it's acting like a ground, not enough current/amps. If it was shorted to a non-fussed positive connection/terminal then possibly. But you would end up with the positive wire catching fire.
I would first look for another heat source. Is the exhaust blowing on it? Does it have excess friction from something, constant rubbing on frame or road?
Try hooking the trailer wire harness to the vehicle, do not hookup to ball. Attach safety cables/chains as usual. Test with driving lights on for awhile, does the cable get hot? Try with signal or brake lights, does the cable get hot? If it doesn't obviously the heat is not electrically caused.
Personally I would recommend 2 things.
1.) Run an actual ground wire from each light and hookup to the trailer plug. Make sure the tow vehicles plug has the ground wire at least attached to a clean contact on the frame or a proper ground wire.
2.) Replace the safety cables with actual chains. The cables are made of multiple thin strains of wire which can rust, fray, wear and break over time. Also they might not be legal on the road in all places.