Well, sorry for your leak, and don't stop ripping out walls. What you have written, doesn't make sense. Whether you are on city water, or using the pump, (and don't be trying to do both at the same time), your whole system is pressurized. If it leaks with one, it will leak with the other.
You wrote, "Water leaks out from the outside of the trailer through the city water hookup". If it is leaking on the outside of the camper, from the city water connection, then the check valve, aka back flow preventor, has gone bad. If it is leaking inside the trailer, could be the caulk around the city water inlet, or could be a bad crimp on one of the lines there. You can pull the city water inlet from outside the trailer, and watch for leaks. Just re caulk when you put it back. Most all the plumbing can be changed without tearing out walls, just be patient, stop, think, and look.
2012 Ford F250 ext. Cab 4x4
2002 Jeep Wrangler Sahara
2008 Yamaha V-Star 650 Classic
2008 Work and Play 18LT