You could try adding a 2x4 or 2x6 to the wall of the closet by using a few wall anchors or toggle bolts. Make sure it's secured firmly to the wall before attempting to mount your TV. A frame inside the closet is probably not necessary, but you may want to consider doing so depending on the weight of your TV. Good luck!

You might want to try using a shellac-based primer like Zinsser Bullseye 1-2-3. It's one of the few primers that can actually block out stubborn stains like tannins in wood, and it's much more effective than latex primers like Kilz. Good luck!