Not sure about seeing any draw if the cap is bad I would suspect not. If the connections are the same at the thermostat you could try swapping the wires around for front and back to eliminate the internal control, after that it looks like you need to get on the roof. Personally I would not suspect bad shore power if one works and one does not but you could check by just turning off the one that works, of course a meter would show low power so you could just use that. I once had one fail where it used what looked like phone line connectors and all I had to do was unplug the connector at the ac end and plug it back in.
My best guess is still the start capacitor as they are notorious for failing.
Good luck and be sure and come back to let us know what it takes to fix it.
