Bad idea.
A governor in Minnesota did that back in the 1970s. When Mondale resigned to be Carter's VP, Wendell Anderson resigned as Governor, and then the Lt. Governor Rudy Perpich succeeded him.
Perpich then appointed Anderson to the Senator's seat. In the subsequent election, Republicans swept both Senate Seats and the Governor's office. It's still known as the "Minnesota Massacre".
I think that Alaska requires a special election, after Murkowski appointed his daughter to a vacancy. And you know how that worked out for him.
Sarah might do well in the special election, but I suspect she would rather stay in Alaska.
You’re right, I had forgotten about that fiasco.