Ok, let's back up a bit:
Converter takes in shore power and puts out 12VDC. Some of that goes to gizmos, some of the 12V
goes to charge batteries. In many (but not all) trailers it is wired directly to the batteries.
Inverter takes 12V
from the batteries and turns it into 120V for off grid use. It will run a TV or a USB charger. It's not going to run the air conditioner. In many trailers the inverter wires directly to the batteries.
What pretty much always is on some sort of relay are the various 12V
loads in the trailer. This includes lights, fridge, fans, and some electronics. The relay is run by a switch that gets called something like "master disconnect".
In some cases people have re-wired things to put in a full disconnect. The most common approach is to use a big switch rather than a relay.
Solar at 75W is quite adequate to handle the various minor loads that are on your battery all the time. It's a great way to keep the battery trickle charged when in outdoor storage.
You can download the manual for your trailer off the airstream site if you do not already have a copy. In the back of it, there is a full schematic showing how the trailer was wired from the factory. That's a pretty good start. Since others have had their fingers in things, it may not be telling the whole story.
Best place to start is with a multimeter. Look at the voltage on the battery. It should never be below 12V. If it is the battery is past the "charge me now" point. If it's down below 11V you likely will hear things disconnecting from the 12V. When on shore power, the converter should get the battery above 13.2V (like to 13.7). If it does not, check the converter wiring. There are other things you can dig into, that should be enough to get you started.