Without knowing more about the TV I can't tell if it's trying to draw more than 100 watts, but this I do know, When figuring amps, you normally divide watts by volts (at least on the DC side, but more complex on AC)
You would think that means 100/12, but when running through an inverter you divide not by 12, but by 10
This means you are drawing 10 amps
That, is quite likely the max rated current on that 12 volt outlet UNLESS it's the one that's on the wall plate with the antenna connection, then it's less.
You need to get a better inverter (There are some nice 300 watt TRUE SINE WAVE jobs out there) I assume your batteries are on the tong, In this case mount the Inverter in a nearby compartment (If the batteries are inside a compartment mount the inverter in the NEXT compartment) connect with starter cable from your favorite auto store, do not forget a fuse, (300 watts will need at least a 40 amp fuse, but read the owner's manual for the proper size) and run 120 volt wire from the inverter to the TV.
Though some folks get along fine with an MSW inverter,, There is a long list of things that MIGHT not work properly with MSW.
With a True or Pure sine wave, that list is.. Well, non existent.
The only ting on it (is also on the other one) "Items needing more power than the inverter can provide" (Which makes sense)
Which is why I recommend TSW inverters./