TV watching with separate inverter?

I'm not 100% sure what you are asking, so here goes with an explanation....that you may or may not know. An typical inverter will need a 12VDC power source to "pull" or draw power from, which in the case of an RV would normally be your battery or batteries, depending on what you have installed in your rig. An inverter would then take that 12VDC power in and convert it to 120VAC. The requirement for the size of the converter would be such that it's output (120VAC) would be enough to supply the television's power requirement....whatever that is that is listed on the tag on the television. And of course nothing is for free, so the amount of wattage into the inverter (read DC volts x amps = wattage) would need to be sufficient. Let's work it out backwards for this example. Let's say the television (TV) draws 3 amps of 120VAC to operate. That equates to 360 watts of power. In order to get 360 watts of power out to the TV, the inverter would have to be supplied with 360 watts of power to supply that amount. And remember, I'm not trying to calculate any of the losses involved in the DC to AC inversion. So to calculate the input power/wattage you would use this formula. 12V (the battery voltage) multiplied by some number = 360Watts of power. So 360 divided by 12 = 30 amps of DC power into the inverter. So in that instance, counting losses incurred in the power transformation, plus any possible inrush current factors when the TV is turned on, I would probably say at a minimum you would want to be looking at a 400W inverter. The DC circuit would need to be able to supply just under 35 amps to the inverter. That would require 10ga wiring from a battery source, possibly 8ga wire, depending on the length of the wire in the circuit from the battery to the inverter.

For the inverter, a small one like 300W or 400W should work fine for a single TV. Keep it as close to the batteries as possible to avoid losing power in the wiring, then just run an extension cord to the TV.
Lithium batteries would give you more usable power (you can use almost 100% of their capacity), so two 100Ah lithiums would easily double your runtime compared to lead-acid.
If you’re curious about how to figure out the amps your TV will draw and want to plan things out better, this page might help: how many amps does a TV use. It’s been super handy for me when setting up my off-grid setup.