Author wants to withdraw or insist on re-reviewing at proof stage of publication
While you might be in the legal right, I think trying to publish when the author wants to withdraw is likely to be more trouble than it is worth for a small society journal.  You probably don't have the resources to get into a protracted battle if the author decides to be really problematic (e.g., a lawsuit).

As I see it, there are three reasonable approaches at this point, in increasing levels of assertiveness on your part:


If the new content isn't very big (e.g., a paragraph or two and the reference), just let it be; just make the change at the proof stage and let it stop being your problem.  If it really does make little difference to the paper, as you say, then this is a reasonable triage to do to get this author out of your hair.
Reject the paper and send it through a new round of peer review.  It's not too bad a waste of resources, particularly if you use the same reviewers, who will be able to just look at the new version in terms of its differences from the old.
Offer the author the choice between a final rejection (no resubmission will be reconsidered) and publishing as-is.


Which you choose probably should be guided by just how willing you are to invest your time and reputation into a battle with this author.  Some people are known and repeat-offending bullies, and it's definitely worth standing up to them.  Other times, you've got a person who is normally reasonable and who has just gotten a particular fixation on this particular issue and it's worth accommodating them one this one occasion.  If you do decide to accommodate the author, however, you need to make it very clear that this is an unusual exception and you will be holding them to much stricter guidelines in the future.