Turn off or disconnect the solar as it's confusing the issue. Unplug shore power and it the battery voltage is above 12.7V add a load (or wait overnight) until the surface charge is depleted and you have 12.7V.
Now plug in shore power and you should see 13V+ at the converter and 12.7 at the battery. Track down where the open circuit is located.
Tip: Measure voltage from a frame ground ONLY and not a ground wire. You might have a ground problem and using a ground wire can mask the problem. I carry a 40' wire with clips for this purpose. Any size wire will do as voltmeters draws are extremely small.
Also consider adding a battery monitor. There are several inexpensive ones available and have been posted on other threads. A AC/DC clamp on amp meter is another good option. From your description you'll see 0A into the battery.
