The issues with running a 12 volt TV on a 12 volt caravan supply are:
1. The caravan supply is never 12 volts - unless the battery is about to die. Normally a fully charged battery will deliver 12.8 or more volts, but in truth that shouldn't be an issue.
2. Most caravan 12 volt power supplies are crap - they supply unregulated voltage to charge the caravan battery - typically around 13.5 to 14.5 volts. May not be an issue - depends on the TV
3. The unregulated power supply in a caravan is designed to run a few lights, or charge a battery. So there is a fair AC component on top of the 12 volt DC.
If your TV doesn't work too well on 12 volts - it's probably the AC ripple from the mains power supply corrupting the "pure" 12 volts.
Either use an inverter to run the TV on mains (a modern solid state inverter is over 95% efficient) or buy a dedicated 12 volt stabilised power supply for a caravan to fit just before the TV - this will clean off the AC ripple
Robert