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Here's a description from the Outback site:
PSX-240 (Autoformer with Housing)
The OutBack PSX-240 (6 kVA) autotransformer can be used for 120 to 240 Vac step-up or 240 to 120 Vac step-down applications, generator balancing, general split-phase output balancing, or balancing the outputs of series-stacked inverters. Incorporating a transformer with 120 Vac / 30 Aac primary and secondary side, a temperature-activated cooling fan and a 25 Aac dual-pole breaker in a steel enclosure, the PSX-240 is ready to install in your custom application. Use for 120/240 Vac, 60 Hz systems only.

PSX-240-Relay (Autoformer with Housing and Integrated Relay)
The OutBack PSX-240-Relay has all the attributes of the PSX-240 and more. Equipped with an internal relay with external controls, the transformer can be disconnected from the system by a simple 12-volt signal. In customized systems, which may alternate between both split-phase and three-phase sources, this protective switch allows your PSX-240-Relay to be easily and safely removed from conflicting voltages. Use for 120 or 240 Vac, 60 Hz systems only.


So it seems this might give you 240Volts to operate your cooktop and dryer and possibly other 240V items off of your inverter?
Don't assume your batteries are bad just yet, running an air conditioner off of batteries and your inverter consumes quite a bit of amps, possibly 100-150 amps. And due to Peukert's effect and inverter inefficiency could be even more. Your 6 8D's have about 700 amps available at 50% discharge. So with new batteries the most you could get would be 5 hours with the air conditioner. But you're running the fridge and other things. So 2 hours could be close.
My thinking is the air conditioner on inverter would be mostly when driving so the batteries are receiving a continuous charge.
A phone call to Newell should tell you their thoughts on this assuming they installed the system and not something the previous owner did.
Either way it sounds like a neat setup.