The typical RV roof AC is an awful load for an inverter to start, even a 2kW one. I don't think a refrigerator with a 300W motor presents the same problem.
Many users do not understand the huge current drawn by a large inverter, the voltage drop due to the wires between the inverter and the battery, and finally how much a lead-acid battery voltage sags under such large currents.
The 1kW Xantrex has a surge capability of 2kW, hence I thought it would be able to handle a 325W motor.
About the auxiliary battery, its function is more to provide power close to the inverter to handle the starting current. To put a battery inside the car, one needs an AGM battery which does not spill acid and discharge hydrogen/oxygen. And AGM battery is head-and-shoulder above deep-cycle marine or golf-cart batteries in terms of handling large currents.
A common misconception is that a 100Ah battery will last 1 hour under a 100A load. No, it's more like a few minutes.
The 100Ah capacity is measured at a 5A load for 20 hours. When the load is multiplied up 20x to 100A, the duration does not reduce by the same factor of 20x to become 1 hour. Under 100A load, the battery capacity becomes more like 10Ah, and the duration is 6 min (1/10 hr).
This effect was observed on lead-acid batteries in 1897 by Peukert. Look up Peukert's law to learn more.
AGM batteries are much better than flooded lead-acid types. The best is lithium battery which is closest to the ideal battery, but its cost can be prohibitive. There are several 12V-compatible LiFePO4 (lithium iron phosphate) batteries on the market now. Look up the price of a 12V 100Ah lithium battery and see for yourself.
I recently tested these lithium batteries last summer, and a 100Ah battery ran a window AC drawing around 500W from a Xantrex 2kW inverter. The current drawn on the battery was 45A at 13V. The AC ran for 2 hours.
PS. Forgot to add, the 2kW Xantrex was cruising under that load. Its fan did not even turn on. The case was barely warm. Increase the load to 1kW+, and it started to get warm and turned its fan on.