the issue you describe with separate bulbs for brake and turn signals on the TV is solved with the 3 to 2 wire adaptor.
The 7 pin connector which is the standard for trailers does not have a brake light wire. Not sure what jayco did or if someone custom wired your 7 pin connector and maybe the trailer. But i'm thinking they had use one of the other pins to make it work? They used either the Brake wire that actuate the electric brakes from the brake controller, the battery wire, or the Aux wire used for backup lights.
History of the issue: All cars use to have only three wires for the rear lights plus ground. The standard 4 pin connector with Running, L-turn, and R-turn and ground. Brakes used both L & R turn.
If turning or 4 ways while braking. Relay logic turns off that turn bulb at a flashing rate. Somewhere back in the 80's/90's the europeans developed a standard for independ turn signals and we started seeing the rear yellow turn signals. This required 4 wires plus ground (5 wire). US followed in the 90 and 2000's and most turn signals went back from yellow to red. But using 4 wires plus ground (5).
Trailers have stayed with the 3 wire plus ground (4 wire) for the lights. But the industry has made an adaptor for the transition from the 5 to 4 wire system. It's called the 3 to 2 wire adaptor, which references how many wires are used for turn and brake lights. I think you need to wire this on the TV 7 pin connector.
