Scott,
I had the same prob. as you, ether worked but nothing else. What shape is your carburetor in? My first guess would be that the carb. needs a good cleaning (only because that's what mine needed). I disconnected the fuel line going to the carb. and directed it to a bucket. Turn the eng. over and see if you get spurts. I know you can't measure the pressure like this but the fact that you just replaced the fuel pump 'should' overshadow that question. I don't know about vacuum or those types of things,
IMO, electric fuel pump is only addressing the SYMPTOM. I'd say you have an intermittent problem with your carb. Get a kit ($13 from NAPA) and completely clean your carb. up. Don't forget to document your needle settings before you take them out.
Rance (Not a mechanic. Not even on TV)