Hey FP,
Your right on top of everything

Yep, UPS is the way to go, I might also add try to find a SMART UPS, most of those are sinewave inverters.
If you want to run the PC while not driving or running the engine you may also want to get a deepcycle battery which would be far better. Car batteries are not meant to be discharge much, just to start the engine then stay full charged. Running them down much and recharging will shorten the life fast and kill them, how long depends how much and low you drain them.
A deepcycle is built heavier and meant to be cycled harder/lower. Still longest battery life is if you only draw it down to about 80%. The deeper you discharge it the shorter life it will have. I use alot of 6V golf cart batteries wired in serries to what ever volts I need, 12V, 24V, 48V for RE type stuff, Inverters and Ups's.
If you go with a deepcycle battery (wall-mart has a decent $55 115amp 12V one) I would also pop for a battery isolator for charging daul batteries. Maybe $25 at Autozone or such auto store. Various sizes and prices. That charges both batteries off your alternator when the engine is running but when you draw power from the extra battery you don't run down the engine battery. Nice to have so you never get stuck with a dead battery and not able to start the car or truck.
I might point out that when figuring watts and amps for such things, Watts Is Watts, but Amps is X 10!
In other words, 350 watts is 350 watts rather 120V or 12V it's still the same watts. BUT at 120Vac that is about 3amps, coming from a 12V battery that is about 30amps! Volts X Amps = Watts. 30amps is alot to pull from one 12V battery, plus you have the wiring and inverter losses so maybe more around 35amps. Even a 115 amp deepcycle battery is only gonna last maybe 3 hours till totally dead, and if you do that much it won't live long. Car batteries won't be nearly as good or give power as long.
Another thing is with batteries amp ratings they are normally figured over about a 20 hour period so in say 20 hours you would get the 115 amps with a slow steady draw of about 6 amps. When you try to suck out 35amps fast for 3 hours you get alot less amps.
Take golf cart batteries for instance. A T105 is rated for 200 amps over a 20 hour period, but in a high fast drain use like golf carts they are rated about 105 amps.
Most of the UPS's I have are 24V and about 1000 watts. I have a big one 3,750watts and it uses 48V.