Had a very similar problem after our water maker was installed. The factory had failed to install a strainer before the inlet to the pump so plastic drill tailings from the water maker installation
worked their way into the pump, causing the discharge check valves (main check + one for each cylinder) to leak. I swapped the pump with an identical pump in the sea water system then disassembled, cleaned, and reassembled the pump. I had to swap pumps several times before I was able to find the correct size strainer for the inlet to the pump.
We were instructed to leave our fresh water system pressurized so that the water maker can flush itself automatically every 5 days. This could obviously be managed by monitoring the flush time interval but it's much easier just to leave the system pressurized. Any unexpected pump operation causes us to search for leaks