Sorry to keep asking you about the "dummy" things, but trying to eliminate the simple & potential external causes before deciding that something is malfunctioning.
1. You have made sure that neither the fridge or water heater is on 120vac power, right? I ask because appliances with thermostats can switch themselves from almost zero to full demand at any time, producing an unexpected overload. No, you need not worry about disconnecting batteries - the charger function rarely consumes more than 1 amp @ 120v.
2. The Overload light on the companion comes on, right? But NOT until the a/c has been running for a few minutes? Then 4 seconds later the Output light goes off? Has Honda explained how that works in conjunction with the Parallel kit? Same, or some higher output level? The standard document says it comes on any time the Companion output exceeds 2200 watts, which doesn't seem likely once the a/c compressor is running. Output should drop to around 1500 watts once the initial surge passes.
3. How about putting a Kill-A-Watt device between the trailer and the genset, so you can see the wattage and such as it happens? Only good for 15A, but good enough for just the a/c.
4. Honda has different carburetor jets that are recommended for 5000 ft and above. You lose low altitude capability, though. Learn about them here:
http://cdn.powerequipment.honda.com/pe/pdf/pubs/pci54675.pdf