Sounds like 12v is not getting power from the battery to the converter. The battery power is connected directly to the converter, and is distributed throughout the TT circuits from there. I'd check the back of the inverter to see if you are getting the 12v from the battery. If not, you have a line break or a blown fuse or breaker somewhere between the battery and the converter.
I'm thinking that it's likely a bad auto-reset breaker (about a 1" x 3/4" little box) mounted very close to the battery. The 12v supply pin from the tow vehicle side is likely taped in beyond that breaker, so that it's not "double fused", and this is why you have 12v when the TT is connected to the tow vehicle.
Similarly, you have 12v when plugged in, because the converter is functioning properly.
The REAL danger with your current situation is that the 12v supply from the tow vehicle WILL NOT provide sufficient power to operate your brakes safely, PLUS, there will be NO power, should the break-away cable have to activate.
2014 RAM 2500 Laramie 4x4 Quad Cab w/Cummins Turbo Diesel, 2009 Rockwood 2607, and 3 'yaks!
300W of Suntech Solar, a Rogue MPT-3024 Controller, and a Xantrex PW2000 Inverter