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How much power will the whole affair suck - do you know? If it's any where near 300 watts then you'll be pulling around 22 amps. That's a lot. Considering that you don't want to go below 50%-odd charge, then you're looking at supplying about 40 minutes before saving the battery from harm. You CANNOT let it run flat - you'll kill it.
I take it that you'll be running heaters too.
Running off the running vehicle is probably not a great idea as the alternator will supply the rig about 14.5 volts while the engine is running. Can the inverter take this? Probably. Drawing 20 continuous amps off the alternator is like running a couple of heavy-duty driving lights and a bit of other gear. If your car only has a 50 amp alternator then it's going to be a bit strained I think - but I could be wrong. If it's a larger car then you're probably running a larger alternator anyway.
In short - not a great idea - especially with the car merely idling.
I run a 100AH deep cycle battery. 29AH is a piddler.
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