I have a created player in my franchise that I play both ways. I am able to DH him any game that he is not the starting pitcher for. So, theoretically, you could DH him 4 out of 5 games. I have had no problem DHing him the day after his start, though I don't normally do that, because of the "stamina" problem.
Basically, if you DH a pitcher, his energy doesn't increase (It doesn't decrease either) So, basically I will rest the player the day after he starts, and the day before he starts, and DH him the other two games. Then, I use the "work around" to allow him to hit in the games he pitches as well (I use the Tigers) So, essentially he hits 3 out of every 5 games played.
The "work around" is this. On days where that player is the starting pitcher, what you do, is put a player at DH who you want playing in the field. Then, start the previous game's starting pitcher, at that position. Then, when you are in the field, go to substitutions, and move the current DH to the position where the pitcher is at. This will eliminate the DH, and force the pitcher to bat.
EDIT: I just saw you were playing The Show 20. Not sure if it works that way in 20....