I tidied up the question a bit :>)
You have a regulator / control box issue. You have two systems;
You have a towable trailer. A vehicle battery in it that gets charged as you drive from a cable in the plug. When you stop that battery ceases to charge. No power in. If you use lights or use an inverter to take the 12 volts and turn it into 110 volts to run appliances, the battery is draining until it becomes flat.
So,..... we bring mains power hook up into the equation.
The mains power should run all your mains 110 volt appliances and provide a trickle charge to keep that battery topped up, but the battery is NOT actually being used, just being charged.
So that is the theory, somewhere you have a piece in the jigsaw puzzle that isn't working. You have a controller box that isn't doing what it is supposed to do. I don't know the trailer, you probably need a dealer that does. Fleetwood went bankrupt in 2009.
If you Google
fleetwood prowler electrical schematic diagram
There are wiring diagrams. I see a battery disconnect solenoid, but all of the dot .it Italian and .fr French sites as scam sites. There are also RV Groups that might be more help than me.
This might help
https://axleaddict.com/rvs/RV-Electrical-Tips-and-Reference-Data
Hope the information helps. A short term solution would be a full size batter charger charging your battery overnight after disconnecting it from the negative lead so just the battery is being topped up. You don't need an inverter if you have mains hook up.
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