The pump is 12v. You're older trailer probably needs the battery hooked up for the convertor to supply 12v.
The only things in your trailer that are 110v (if you have them) are: the outlets, the Air Conditioner, a Gas/Electric fridge, and the microwave.
Everything else, the lights, the water pump, the vent fans, the furnace, are all 12v.
The A/C and the Refrigerator also require 12v for the controls.
However, you say you are on A/C and everything is in working order. Does that mean that the lights are working? How about the fridge? Vent fans?
2004 Dodge Ram 2500 Club Cab 4x4 CTD
2007 Sunline Solaris T-276-SR: Misty Blue/Cherry
2001 Dodge Ram 1500 Reg Cab 4x4 OFF-ROAD