Game situation was: Stunned Pilot crit after moving over asteroid and finishing on asteroid. Did the ship need to take the damage as finished overlaping?

I'd say no.

The Stunned Pilot card goes into effect when you overlap an obstacle, and at the point you got it in that case you had already started the process for that overlap. So I'd say that the trigger has already passed.

Nope. You roll for asteroid damage after you finished the maneuver and skipped your action, so you do not yet have the Stunned Pilot card at the time you hit the second asteroid.

'ed. Twice