Liberty XL2 has the Continetal IOF 240 with FADEC. Would the engine run if
the batteries failed? It does not have a magneto. Would the ignition system
run off of the alternator? The salesman at Sun-n-fun had no clue. Does
anyone know?

I would think that it would run for a little while but an alternator needs a
source of electricity for the field and this comes from the battery. There
must be some sort of redundancy in the ignition system. I would think there is
a second smaller battery intended to allow the system to run in the event of a
main battery failure. Can't see it being certified otherwise.