bad battery, system wont work on 120AC

Our rig works just fine with the battery turned off and running 12v things through the converter while on shore power. Not sure if a bad battery would keep a converter from operating properly.

Some converters don't operate properly without a battery. It's very reasonable to assume that a bad battery could also cause improper operation or non-operation, depending on the battery fault.

Connecting most any known good 12V lead acid battery in temporarily would let you check the converter using a voltmeter. Even something as simple as connecting the trailer electrical system to the truck battery with jumper cables (carefully) should be sufficient to verify the converter output voltage.

If the battery has an internal short it could cause the converter to trip constantly. I'd remove the battery negative cable, rest the converter and see if it powers up. If so, the battery is the problem.

The first thought is that the converter is bad. But that is after all connections have been checked.
If the converter is bad....or bad connection, could that make you think the battery is bad, when really it is the converter not charging ??

If you had a simple thing like a voltmeter, all your questions would be answered

If the battery is removed, it may be that the converter was disconnected as it first attached to the battery. Or, it could be that the converter needs to see something close to turn on at all.
If you come back with a converter model, someone might know.

Matt