How are emergencies handled during In-Seat Rest?
Option 2:


  the pilot flying continues to fly while the pilot not flying
  immediately goes through the engine out checklist, despite the risk of
  sleep inertia.


When waking up to an emergency situation your body will start to produce adrenaline which gets you over the sleep inertia very quickly. If the situation is not clear to the PNF, he will be briefed by the PF. 

Besides that, in a checklist flow, both pilots are involved and crosscheck each other's actions.