I'm not familar with the particular trailer set-up, but some electrical systems run from the battery circuit always, and the inverter/charger serve the role to keep the battery up. They are configured this way, so that in the event of a shore supply outage, there is no real switch over to battery, because you're always on battery, so to speak.
So, that corroded fuse holder was adding resistance, resulting in the voltage drop under load. Likely the inverter/charger is on the battery side of that particular fuse, not the load side.
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2013 Jayco Eagle 31.5RLDS
2018 Silverado 2500 HD Duramax
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