I'm going to join the discussion here because basically I have exactly the same problem (or at least the consequences of a similar problem). Even though the car has had an engine transplant and an aftermarket ECU I don't understand why you need another relay when you should have an EFI Main Relay (like in the photo). The only answer seems to be that the conversion shop went with the standard non-EFI relay and then just added a bunch of extra wires to complete the conversion to EFI. Always a recipe for disaster, especially as the wiring looks like its been done with crimp connections, notorious for working their way loose over time.

The first thing is always to test the fuel pump is actually working. As you have found out, and others have suggested, a simple 12V feed to the correctly grounded pump will tell you if it is working. So far so good.

Your main issue is with power and you seem to have solved that, at least temporarily, with some new wiring. But I wonder if you are getting VTEC engagement as that is a function, in some ways, of revs. My understanding is that the fuel pump needs to be switched via the EFI Main Relay, and communicating that to the ECU. I'd be interested to know your final solution.

Peter