I would say that, as the cultist attacks, Harvey can either fight back, dodge, or maneuver. As drawing the gun and shooting is an attack, it is not an option at that time.
On Harvey's initiative, he can draw the gun and fire. (Debatable if this is an other action or not, I'd say not if readily available from holster or pocket) Firing into melee is at disadvantage, but point blank is advantage - they should cancel out. (I could be convinced that as Harvey is in the melee, he is still at disadvantage)
Next round, Harvey will act first as firearms are at dex + 50. The cultist is in trouble as he brought his fist to a gun fight!