kind of OT: any suggestions for 12V to 110VAC inverter?

Necessary, although you need "peak power" for long enough to supply the surge -- if the "peak wattage" specification has become a victim of specsmanship then someone will figure out a way to make it huge and meaningless.

Because cigarette lighter sockets are limited in current delivering ability. Get the inverter you need, then if necessary wire a connector into your car that will carry the necessary current for the inverter.

Good luck. RV magazines would probably be the best place to look for reviews.

Find a small UPS that someone has discarded because of a bad battery.

Cigarette lighter (I think the politically correct have managed to rename this a "power port" for fear that calling it a "cigarette lighter" will encourage people to start smoking which, in turn, will lead to harder drugs, drunk driving, etc.) is usually only rated 15A (maybe 20A). So, you have less than

You have to deal with intial turn on surge as well as when battery voltage sags. If you can't count on a still 12V supply, the UPS will yell at you (and possibly shutdown).

Better bet is to add a dedicated wiring harness right to the battery so you can cut down on I.R losses.

Is this a CRT TV ?

The starting current can be quite large due to the degauss coil.

Take a look what a new TV that operates directly from 12 V would cost, since you would avoid the power losses on in both the inverter and the TV power supply.