I'll keep this short, since I know you're busy. But the only reasonable lineup against a RH starter for the Yankees (Severino or Tanaka) involves batting Matt Joyce in the top third of the lineup, in the catcher's position.
The roster is likely to have one LHH bench bat in a game against a RH starter, and there's no guarantee that player would even get an opportunity to hit. The Yankees have two high leverage LH relievers in Aroldis Chapman and Zach Britton. Additionally, the standard lineup against a RH starter would look like this:
The bench would be:
As such, we have one lefty bat who may or may not get in the game. Additionally, none of the right handed bats would justify being pinch-hit for.... except Jonathan Lucroy.
By batting Matt Joyce in the first inning, we assure him an opportunity to hit against a RHP, we limit the number of plate appearances by your worst hitter.... And we assure the fact that we can pinch hit for the catcher later in the game. If you pinch hit for Lucroy before he enters the game, you still have two catchers eligible after the top of the first. This allows you to essentially pinch hit for Lucroy twice if need be -- once in the top of the first inning, and again later if, say, his spot comes up against Aroldis Chapman in the 8th or 9th inning and Mark Canha is still available. You can bring in the stronger bat and use Josh Phegley defensively.
This strategy would guarantee an at bat for your exclusive RH-mashing hitter without defensive value, limit your exposure to having a weak hitter forced to remain in the game late, and efficiently utilizes all of your position players by maximizing Phegley's potential usage.