Thanks for the replies. Friday after work I'll fill the fresh tank and pressurize the system and see what I see.
Correct me if I'm wrong but if the system is pressurized and the pump turns on for a second for no obvious reason wouldn't the leak be past the water pump... Meaning the leak is in either the hot or cold water lines inside or the hot water heater? Or could it turn on too if there was a leak from the fresh water tank to the pump? I looked inside today by the water pump, under the shower, and by the hot water heater with no evidence of water inside. It's pretty dusty in those places so if there was a stream of water I think I would have seen something, but who knows.
I think having the heated tanks and enclosed underbelly is nice, but not seeing what is happening underneath is driving me crazy.