OK, there are a couple of things going on here. Leaving the bypass valve open will definitely cause tepid hot water (cold water will mix with hot when the hot water faucet is opened) but it will not cause the OP's pump run issue.
I am still sticking to an air leak in the pump or earlier part of the system and the pump is losing its prime.
A bad diaphragm in the pump could do it as DunnNC said. Or an air leak anywhere between the pump and the fresh water tank.
Lou and Laura with Bella - German Short Hair Pointer
2008 GMC Sierra 2500HD Crewcab SB Allison Duramax
2010 Flagstaff 8526RLWS - Superglide 3300
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