I guess ill have to read up on the carb. Is it not vented? If so too much fuel would run out the vent. Gas washing the cylinder should only be able if you oversized the jet. The jet plus maybe needle adjustment should set up max fuel in the cylinder. Pressure in the bowl should not effect how much fuel goes in the cylinder unless it is way off. I could see more trouble if this was a diaphragm type carb but it's not, it's a float type.
As I understand the fuel relay, when the ignitor is on/running, the relay is on. No mater what pump you use, 12 volts is 12 volts. I did however order after market relay for 16 bucks. I'll run with out one before I spend 90 bucks for something I really don't need. I don't know a reason to have the ignition on and not actually start the engine.
As of now it seems to be running great, a little cold natured to start. Guessing if it was running rich/ gas washing the cylinder it would start up much easier, as if the choke is closed. fr
With all that said I could be wrong, I was once. lol Feel free to correct me. The biggest thing I have learned in life is you are never always correct and you can always learn something.