You can install a fuel pressure gage in the fuel line, mechanical T should be ok if you can see the gage while engine is running, otherwise plumb in an electric sender and put another gauge on your dash!

Fuel pressure should maintain at a steady 4-7psi for a carbureted engine...if it drops the fuel pump is not maintaining enough flow to keep the float bowl filled...the float drops and the needle comes off the seat but not enough fuel gozinta the carb and the engine stalls.

If the float bowl fills up fast enough the engine will catch and keep running.

I guess another problem could be an obstructed needle seat and gasket (you said it was recently rebuilt...did this 4K cutout problem start after the carb rebuild?)