I would start by looking at the battery circuit. While plugged in to shore power the converter will provide 12V to DC panel, when unplugged it appears the battery is unable to provide 12V.
Use Dan's trouble shooting link in his post above.
You mention the battery is good - have you load tested the battery? There are cheap battery load testers to be had a parts stores. The battery may show good voltage when plugged in, but you should disconnect the battery, let it "rest" for 30 minutes and then test voltage.
My thought is the battery connection is bad, or loose connection on the positive wiring or negative wiring, preventing a closed circuit - check the connection to the frame as well - follow the negative battery cable back to where it is grounded on the trailer frame.
Hope this helps,
Brian