As your new pump hasn't fixed the fault and doesn't have a pressure cut-off switch, I'd stick the old one back on.
Maybe it's an electrical problem, intermittent fuel pump relay perhaps.
The control signal for the fuel pump relay comes from the ignitor unit.
Try the usual search for loose and corroded connectors, damaged wires etc. and if nothing looks wrong, a new relay.
<edit> Sorry. Ignore this, missed the post where you found the original problem is solved!