If the key is in the on position, engine off, pushing the flap open (just a little bit) should cause the fuel pump to turn on.
Engine off and doing this you should be able to hear it come on.
(Engine running, the pump should be on -> airflow opening the flap.)
If it's not coming on you might check the double relay for power being supplied.
Here's another check that is dear to my heart:
You might pull the electrical connector off the AFM and check across the "switch" for a clean contact when the flap is pushed open a bit.
I've fixed a stalling condition before by cleaning the "switch" contacts in the AFM, and I'll be doing it again soon ...
And as always, before getting too far out in the weeds, refer to the FI troubleshooting guide for an orderly approach.
(The short run time, then dying thing might be explainable by this ... cranking turns the fp on ... engine starts, airflow starts, the flap should open and keep the fp on when you release the key to the run position. If the power is not there ... it should sputter off as the available go juice is consumed.)