I know its been awhile but the clicking is more than likely the ignitor trying to ignite than the AC unit? Since you said you heard this when you also ran out of propane, this is what I would expect.
I can't imagine it a result of a low battery but i am making the assumption your connected to shore power.
I'm wondering if there is a dirty sensor somewhere. Or if there isn't some recurrent air bubble in the line somehow, maybe its caused by higher day temps and colder night temps?
I would find the furthest thing away from your propane tanks (usually an outside connection for a grill) and try to ignite that. Make sure you have good flow to that point (grill works first try, etc), and see if the problem persists. Maybe you have a faulty automatic changeover switch and when it gets cold something happens?
To me though, it sounds like the furnace stops getting propane at some point in the night and starts trying to ignite, which doesn't work until you turn it off and back on. Maybe its an error on a control board or something, and that action clears it?