Check your ground, not for the pump but for the relay module..
I would consider a wiring diagram and jump the connection at the module to see if the wiring going to pump is not damaged. I`m not a big fan of the factory modules myself, much of the time the solder breaks down and they fail.. I`d wire in a standard 4-pole relay, there way cheaper..
Good call on the bypass, most probably would of assumed the pump was bad.