What you are saying is legal under OBR but illegal under NFHS. In your scenario, under pro rules, the role of the DH is terminated and the manager must indicate to the umpire in which slots the SP and the new pitcher are batting. One would be in the DH spot and one will be in the left fielders spot.
Under NFHS, the DH is batting for a player, not a position. If the DH is batting for the pitcher, and the pitcher goes to left field, you still have the DH batting for that person.
Under NFHS the DH and the player he is batting for are both locked into one spot in the batting order. It doesn't matter which team is at bat. The only person the defensive player can bat for is the DH, which would terminate the role of the DH.
Sounds like you are confusing the Pro rule with the NFHS Rule.