This was new to me

[mbyron]

Yes, coach may use a courtesy runner, but the role of DH is eliminated by pinch running the defensive player (the pitcher in this case) for the DH.

Rule 3-1-4:

The role of the designated hitter is terminated for the remainder of the game when:
a. the defensive player, or any previous defensive player for whom the designated hitter batted, subsequently bats, pinch-hits or pinch-runs for the designated hitter