Hi Banjoman,
I tried to find a manual on your model converter, and not having much luck. They have been discontinued long ago.
Many of the really old converters had an internal relay that would connect the battery when the battery was needed. That said, if yours has the relay feature since part of the 12 VDC system works, there are high odds a relay would be working.
Since your 12 VDC lights work on shore power (120 VAC), this points to a board-level problem in the converter on that specific circuit. Since the lights work when the system creates 12 VDC power when on shore power, the lighting circuit wiring and fusing back to the power converter itself appears to be working. The problem is then within the DC control portion of the old converter used to switch the power to the battery.
In the late 70s and early 80s, I worked with hard-wire solid-state control systems for machine automation for my job. The days of AND/OR gates and JK Flip Flops, using TTL logic with "wire wrapped" connections on the back of the master rack for connections for those who remember that era. If you have a buddy or yourself who knows older printed circuit board design and troubleshooting, they may be able to test and back into which board component failed. That would tell you where the problem is. Then, it comes to the hunt to find the part to replace. It is a lot harder than it used to be to find these old solid-state parts pending what they are in low quantities. The days of running down to the local Radio Shack, who did sell ones and twos of many board-level parts, are long gone.
You may be into upgrading your power converter to correct the problem. This is a common thing to do on older power converters, and you gain the benefits of modern-day power converters. Many of the older power converters where not very good battery chargers or maintainers. They were nick-named battery killers as they had no true float mode once the battery reached full charge. They just kept pumping in too much charging power, boiling out the battery, and then early battery death came.
For a new system, I would recommend that you give this place a call. Ask for Randy, the owner. I have bought several power converters from them to replace older Sunline power converters that died.
https://www.bestconverter.com/
You will see that their website has many power converter brands to help you get closer to fitting your upgrade. Sunline used many different brands of power converters as the years went by, and the technology kept changing and replacing the prior ones. Many brands went out of business along the way, forcing a change.
I understand that you're facing an issue with your power converter, and I hope this information can provide some direction in resolving it. Please don't hesitate to share any findings you have, as this can be beneficial for others who might encounter the same problem. Your contribution to the community is greatly appreciated.
John