#1 - I don't think you'd ever really see pitchers hit. The "NL strategy" thing is only a thing late in games. Early in the games it's basically a free out. So you keep the late game strategy, while removing the early game pitcher hitting.
#2 - That is the one issue I can see. However, you are talking, what, 2 ABs a game? So you just have to start moving guys around early, as NL teams have always done when a pitcher got shelled. I would also allow the DH to be shifted, so you could move, for example, the DH to 1B, and have the relief pitcher sub in for the 1B