The Champ should run your 13.5 AC, mine does even at high altitude UNLESS there is a big draw like the RVs converter charging the battery. You said the battery is dead so the converter will draw a lot of power for a while. After a while it will slow down as the battery gets closer to 90% charged and your AC will run. You could also shut off the converter.
Was the battery charged when you bought the rig? If so did you disconnect it or charged it sense? Even just sitting a battery will be drained by the alarms.
You can plug in the trailer or remove the battery and put it on a stand alone auto charger. Take it to a auto parts store if you don't know if it is good and they will test is.