You say that if you hold the tank up (using gravity to feed the pump) it'll start and run but if you let it sit and try to restart it won't push enough fuel.
I'm assuming (yeah, I know!) when you say "push" you mean push fuel from the pump to the carb and when you say "let it sit" you mean returning the tank to it's normal level.
If the line from the tank to the pump remains full but you can't get fuel from the pump to the carb, it sounds like a weak pump.
If your carb bowl is full it should start no matter where the tank is and even if there's no fuel flowing to it, it should run until what's in the bowl is used - that should be plenty of time for the pump to "pick up the prime" and push more fuel to the carb.
Once you get it running let it run for a minute or two to make sure all your lines are full.
When you shut it off, close the fitting on the tank - that will hold fuel in the line to the pump.
At this point you should have all your lines and the carb full and the fuel in the line from the tank to the pump should stay full as it can't drain back into the tank with the fitting closed.
If you try to restart it and fuel won't go from the pump to the carb, I'd personally spend a few bucks and get a Walbro or Mikuni pump. It'll cost more than the one you have but it sounds to me like the one you have ain't gettin' the job done.
Unsaid in all this is I'm also assuming that there are no leaks in the pulse line from the valve cover to the pump - if there are, it doesn't matter what kind of pulse pump you have, it's not going to work.