While holding the mechanical pump(bolts not tightened) to the engine block and having someone turn the engine over I can barely feel the cam touch the pump.
There you go - you found the problem...
Don't know about the 250, but 235 fuel pumps over the years had different lever angles, matching different cams and engine blocks.
I have found 4 different ones for the late 40s-62 235s so far.
Some pumps have such a wide angle ending up too close to the cam lobe, that they will eventually fail after a few hundred miles of pounding the delicate mechanical parts inside too hard.
The smaller angle lever pumps will pump less or barely any gas and the others can flood your carbs, without an additional regulator.
The bummer is, you just can't go by vendors having the right pump for the right engine block year or whatever replaced cam may be installed.
Most sold are remanufactured and they can end up having different lever angles from different pumps.
It's a matter of luck, experimentation and comparison to end up with the right pump.
Took me 4 tries and two broken new pumps (of course both times being farthest away from my garage...) to finally end up with one that delivers just the right amount of gas - not too much - not too little.
! Careful when buying NOS fuel pumps - Just like with NOS carbs, master and break cylinders - they need rebuilding with fresh rebuilt kits.