Can I move the fairie on a bonus turn due to the builder?
A player may move the fairy instead of placing a follower.

A player can move the fairy on the placement of the first tile, the second tile, or both.

A player can only score the one point when the turn begins.  The "double turn" is a continuation of the turn, not the beginning of a new turn.  Therefore a player cannot move the fairy on the first tile and then score the one point.