try a squirt of starting fluid immediately - when it's running rough? at least we'd know if it was fuel-related...?
or, next time you feel it would fail, do a few OFF to ON (not START) key cycles. wait for a 3 count at ON, then go to OFF, back to ON, 3 count, etc. - quickly. That runs the fuel pump and should fully pressurize the rails....?
you said you can detect a little air movement...wonder what would happen NOT at fueling, but say right now you removed and replaced the gas cap? would it start smoothly?