4303 and starting issues
The check valve may help. Check out peterson fluid systems. If they have a check valve, it'll be one of the best you can buy.
Another solution is to wire the pump up on its own switch. That way the pump does not turn off between key on and key start.
The above is assuming you have a perfect start up tune.
The pump will prime as you key on, and as soon as the engine starts rotating, the pump will also start again...ie during cranking ( assuming you've wired it correctly )
I've run 044's for 10 years on mine, they dont maintain line pressure, and I've never had issues starting.

