Well it can only be that you've wired the fuel pump relay incorrectly really. Thats the only thing thats changed between it running and not running...
If you fit a standard relay (generic or nick one of the yellow relays out the fusebox) and pop it in the holder in place of the fuel pump relay, then the fuel pump should permanently run again. See if it does and the engine continues to run etc.

If it does then you know the engine/fuelling is fine and its simply the relay setup.