You will have to upgrade the tiny wire in your TV and TT's wiring harness to actually get 30 amps back there. Even then, you will still not have enough. You will have other loads in the TT like the radio, LP leak detector, etc. drawing additional power. If your batteries were down at all from camping there won't be anything left to charge them.
There's no reason not to run it on propane, that's the normal way of doing things. It's designed for that. It will likely cool better and you don't have to change or buy anything.
