The reason for the failure on the briggs engine is irrelevant at this time.
I swapped engines complete... Carburetor and all.
The question... How can I get the pump engine (and carburetor) to work on my mower that has a fuel tank above the engine?
IF the pump engine has the pump carb on it then you just hook the fuel line to the pump inlet.
The position of the tank will make no difference.
The two spigots on the side of the pump are the inlet & outlet the one in the middle is the impulse line.
The spigot nearest the impulse spigot is the inlet.
The lonely one all by itself is the outlet.
There is a small vent in the pump with a sintered brass filter in it these can also get clogged and stop the pump working properly.
The outlet should pump fuel at least 4" higher than the top of the tank when the engine is cranking.
You might need to dissasemble the carb and clean it sounds like you have some old gummy fuel in there.
Nice new fuel lines would not hurt either same for the filter.
OTOH no reason to assume that the $ 100 JD fuel pump is actually working to start with.
If it has been left without any tubes on the end small insects seem to like to take residence in there.
The head on the pump will make no difference the pump is a simple diaphragm with a light spring on it.
Crankcase pressure causes the diaphragm to move in & out as the piston goes up & down.
The pressure of the fuel delivery from the pump will not be affected by any head on the supply side
There is a one way valve on the outlet that regulates supply pressure and pump carbs have a stronger closing force on the float needle to account for the extra pressure from the pump.