Sorry to hear the situation. Also not sure what your TV is but have you checked the TV owners manual for its recommendations on running with the spare tire? I've never looked at this question as my TV has a full size, but wonder what the load rating of the spare is vs. full size.
Good luck and be safe.
2005 Dodge Ram 2500 5.9L 6 Speed
16" Michelins, Hi Spec Wheels, Max Brake, Carslile Actuator, Equal-i-zer, Dill TPMS. Campfire cook.