I think you will need to figure out if it's losing fuel pressure, or spark. Ima thinking it's fuel, so I'd be putting a pressure gauge on it.
It kinda sounds like the pump loses electrical power in the run position.
I put a volt meter on the fuel pump connections and when it initially primes I get voltage and then then cranking and starting I get my 12volts and then when it dies I lose voltage. I don't have a pressure gauge but I'll have to buy one and install it in the line somewhere as it sounds like a good thing to be able to inspect when needed. The fact that it starts and then dies makes me think it might not be the fuel pump and perhaps an electrical issue? If it was just cranking then perhaps I'd suspect a dodgy fuel pump.
