Welcome Aboard!
Lots of questions and thoughts, sounds like your batteries might have been discharged to a point that the inverter did a system shutdown.
What do you have for batteries? Just the simple group 24 dual purpose battery that came with the TT?
What size inverter are you using? What were you powering off the inverter?
I agree, most likely you just drained the batteries to a point that the inverter shut down. A good multimeter is your friend. While learning your rig, it is well worth noting what you have been using and how long to discharge the battery. This will give you an idea how long you can run things before you need to recharge. Or helps you determine how big of a battery bank you need.
Battery maintenance is important. If you have the cheap battery that came with the camper, they require a little maintenance. Basically check the fluid levels once in a while, and top them off as needed. ONLY use DISTILLED water, nothing else, or you will dramatically shorten the battery life.
Don't know what battery you have. What every the amp hour listed on the lead cell battery, you can draw about 50% of that out of the battery, before it is considered discharged. So if your battery is listed at 64 amps, you can use about 30 amps. So lets say all of your led lamps in your TT pull, say 1.5 amps, and you leave the lights on. Your battery will be discharged in 20 hours (30/1.5=20hrs). I would not be surprised your inverter was pulling 15 amps. So 30/15=2 hours, which seems about right.
I do not follow RogerR's rule of don't discharge below 12.5 volts. I try never to drop below 12.1 volts. I have many times, been below 12 volts. With good maintenance and care, you can get 8+ years out of a cheap battery. I hit 8.5 years with my old set of batteries in my current TT. Last boat battery made it about 4 years. It was not treated as well.
Inverters waste a lot of power, converting from 12VDC to 120VAC. So having an inverter sized correctly for the load is wise.
Being everything worked with the generator, does not surprise me. A completely dead battery will take many hours to recharge. For me, with my 100+ amp hr battery, it takes about 2 hours to charge back up from 12.2 volts to 12.7 volts when on shore power or generator.
Hopes this helps
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