Missions have to be completed at the end of your turn.
I don't see that in the rules. Where did you read that?
Do I have to end my movement in a laser hazard to finish this mission?
I would say yes.
So I would have to just fly in circles hoping to roll the right amount?
The rules say:
Standard Rules wrote:
To complete a mission, a player must land on a surface city, spaceport, or other indicated space in the designated system
The problematic word here is "land", since the rules do not allow a ship to "land" in spaces which are not Cities, Spaceports or Asteroids. Moreover, ending your movement because you are out of movement points is not considered as "landing", but just as "ending your movement". Thus, it is literally impossible to complete the mission you are talking about according to the rules.
Therefore, my interpretation is that these type of missions, unconventionally allow a ship to intentionally stop or "land" (ending your movement phase) on these hazard spaces for the purpose of completing the mission.