Chris,
This sounds like a plugged fuel tank vent. When the engine starts, the pump
is pulling fuel out of the tank faster than the vent can admit air to
relieve the vacuum that is forming. Eventually the pump cannot overcome the
vacuum and fuel flow stops, killing the engine. During the time you are
attempting to fix the problem, some air gets into the tank to relieve the
vacuum enough for the pump to again move fuel to the engine.. until the
vacuum again gets too great and the engine stops. This would be easy to
check. Take the fuel cap off and see if the engine runs ok.
Don (MI)