You may be able to disassemble the pump head and clean any debris out of the check valve. SHURflo pumps come apart pretty easily, but I don't know about other brands.
A water pressure regulator can reduce a higher pressure, but cannot boost a low pressure of course. Does the pressure at the shower increase when the onboard pump is turned on? We've been at a number of parks over the years where we routinely turned on the pump and drew from the onboard tank to get the maximum pressure from the shower. There's no need to turn off the city water when doing so.