Should Shohei Ohtani bat in AL parks on his pitching day, then move to right?

Shohei Ohtani could bat for himself when scheduled to pitch. His team, the Angels, would not have the designated hitter (DH) in such a game.

Yesterday at home against the Minnesota the Angels scored a run in the bottom of the 9th to win 2-1. Because Ohtani was the starting pitcher first baseman Albert Pujols started at DH, third baseman Luis Valbuena replaced Pujols at first and Zack Cozart backfilled for Valbuena at third.

Pujols: single and BB
Valbuena: single
Cozart: 3 singles

For the season:
Pujols (at least 38 years old): 162 PA, 95 OPS+
Valbuena (32): 118 PA, 85 OPS+
Cozart (32): 160 PS, 103 OPS+
Ohtani (23): 74 PA, 181 OPS+

So when the opportunity presents itself, should the Angels forego the DH and let starting pitcher Ohtani bat for himself? Yesterday Ohtani pitched into the 7th inning. By then Pujols, Valbuena and Cozart had already batted three times each. Cozart singled in the bottom of the 7th and singled in the winning run in the 9th.

So, Ohtani would have had three PA  before being relieved. Plus, Ohtani could have remained in the game by moving from pitcher to right field to replace Cole Calhoun, who is hitting like a bad hitting pitcher: 136 PA, 8 OPS+. The relief pitcher would bat in Calhoun's spot in the lineup, which oddly was number 2.

Calhoun has almost twice as many PA as Ohtani. Why doesn't Ohtani replace Calhoun completely on Ohtani's non-pitching days?

When Ohtani eventually pitches in a NL park where there is no DH, Ohtani would, of course, bat for himself. But when Ohtani is relieved, the Angels should do a double switch and move Ohtani to RF, replacing Calhoun.