How do I deal with a stubborn co-author who won't respond to suggestions?
Rewriting the text doesn't necessarily qualify someone to be the first author of a publication. If you're using the intended first author's structural outline and just rewriting sentences, then I don't know if that's sufficient to usurp the intended first author's claim. If I were the PI, I don't think I would do that.

What I might do is place a note in the paper along the lines of "X and Y contributed substantially to the writing of this paper," where X is the intended first author and Y is the person who rewrote the paper.

Better still would be for the PI to work with the intended first author about the corrections that are needed, rather than just rewrite everything. Helping X to become a better writer will reduce the PI's workload in the long run.