The FAQ explains how this works on p24 of the current rules, but in the context of Library Access and Wild Wormhole.
Gain bonus Æmber from card (doesn't apply to Chota Hazri)
Powers that trigger each time you play a type of card (Hunting Witch)
Play effect of the card (forge a key)
After enters play effects (e.g. 3x Autocannon kill Chota Hazri)
To be precise, Play: abilities and "after play" (such as Autocannon) trigger at the same time so active player decides resolution of those. That's because Play: abilities are defined in the rulebook as triggering "after a card enters play" - same as the wording on Autocannon for example.
But anyway, yes, you can forge a key in your situation. Not because you can choose the order, but because abilities that trigger "each time a card is played" (such as Hunting Witch and Library Access) always resolve before Play: or any other "after a card enters play" abilities.