Although don't know the details, the 50A to 30A shouldn't be a factor. 50A RV service is actually (2) 50A legs. To get a single 30A circuit requires replacing the receptacle and utilizing one leg of the power reducing the breaker to 30A. In other words, the wiring should be MORE than capable.
Assuming that the TV is shutting down because of a voltage drop, it may be time to break out the meter and start measuring. I would first start with the TV receptacle, and measure the voltage with the A/C off, then on. Then from there make a similar measurement at the RV's breaker (for the TV) and finally at the source.
Depending on the voltage at the TV outlet, check the television's manual to see if the power parameters are stated. Since you have a 2014, it should be under warranty. If the last measurement (your home power) indicates sufficient power, then it may be time to visit the dealer to see if the shore power cord, or internals may be defective? That's all I can think of.