How would you handle?
To follow up with what Justin said -
Both the target for Den Protector's ability and Deathmists' Reanimate trigger are added to the stack at the same time, in the order that AP choses. When Den Protector's trigger goes on the stack, it must have a target.
There is some OOOS going on here, and I would explain that to the players so that both have a better understanding of why this situation has become unclear. Then back up the play to the DP's megamorph, having him declare a target for DP and announce Deathmist's trigger.
NAP can chose to play Hallowed Moonlight if he wants, but it's worth noting that Deathmist Raptor's trigger is a may ability, and AP can chose to leave it in the GY if they want. If AP chooses to resolve the Raptor's trigger and exile the Raptor, I would not interfere. I would explain the play mistake after the round has completed so that the player has better understanding of how their cards work.