Will shore power run equipment when battery is dead?

When you say nothing will turn on, do you mean the 120v things too (fridge on AC and the microwave eg, or just the DC things?

If the 120v things work, that should include the converter, which should run the DC things with no battery.

If the MW works but the receptacles don't, you might have water in the outside receptacle that will pop the GFCI, which might also stop the converter if it is a plug in type.

So need more info!