If I were you, I'd probably try to make the smallest revisions possible that allow me to state, in the reply to reviewers, that I addressed at least some of the comments of the new reviewer. Sometimes things look big but can be addressed in a small way, for example a reviewer might say "rewrite Section X and in particular change the claim that XX", and you can address that changing said claim only. Whenever they are not specific, there's a good chance getting away with not changing anything, or changing some wordings just cosmetically. I'd also give arguments why I don't do all of them (in case I don't). Maybe I'd add in my comment/letter to the editor a remark about how the process has gone up to now and that demands of previous reviewers have been satisfied, so the opinion of the new reviewer asking for a large rewrite is apparently not shared by the others, and it's not my own opinion either that this would benefit the paper, however I tried to address raised issues as well as I could.
Obviously I can't know to what extent this strategy works here without much effort (for sure it takes a bit), but my impression is that your chances may be best if you continue to "play the game" in the regular way at the surface, still giving the new reviewer as little as one can just about get away with. At the end of the day it will depend on the editor to what extent they rest with the reviewer or with your point of view, and we can't know that. However, if I am editor and I bring in a new reviewer because a previous one dropped out, the new reviewer doesn't necessarily make me happy with a review like this if otherwise the paper looked close to ready. I may well accept in the next round if the author convinces me they have done all that they reasonably could leaving unchanged some bits of the paper that were unfairly attacked. The editor will make the decision, not the new reviewer, and the editor is not obliged to insist on whatever the new reviewer says.