I've thought about it and this was my theory:
As soon as Trench Run is applied to the Death Star Dial (DSD), Trench Run itself becomes an objective like The Heart of the Empire objective (10 damage and LS wins the game). Trench Run does not change the DSD into a damage dial, else the card will just be overpowered; LS could just slap the card down when the DSD is at 11 and win the game.
Thematically, this makes more sense. It wasn't easy to make a trench run, but it was possible hence the 10 damage capability. It also makes sense that it can only be engaged during the Conflict phase ("You may engage the DSD as though it were a DS objective") since it was that way in the movie.
I...think this is how the card works.