more on the C PAP and the 12 volt battery
So, I'm on my third CPAP machine and all of them would plug directly into 12 volts. that has to be an electrical economy.
but still, there is the battery. if the lights dim when the water pump comes on, either the battery is weak or there's a grounding problem somewhere in your circuit, or should I say lack of ground. of course, there are thousands of possibilities for other electrical short circuits to be causing the problem. you can have a load test done on the battery to see if it is holding up. A a highly sensitive, preferably analog, voltmeter trapped in somewhere downstream from the battery will probably tell you everything you need to know. either the inverter is creating a load that is pulling the battery down, so maybe a bad inverter. Or battery weak or not well grounded. You could take the inverter and plug it into the cigarette lighter in your car to test it. those things will labor pretty hard if they're not getting adequate voltage.
just a word about charging the battery. That is not to assume the converter in the trailer charges it well. If it was not maintained well on a high-quality float charger during the offseason the battery is probably toast. the load test will show you that in nothing flat.