2009 fleetwood providence have hot water with city water hook up but none with the 12 volt pump but do have cold with each and plenty of pressure with both
Check that the outside faucet in the wet bay is shut off on both hot and cold sides. That faucet is notorious for allowing the hot and cold waters to mix. This is just a guess that you may be getting higher pressure when hooked to city water, and the mix at the faucet would be less of an issue than when running off the pump.