I just wanted to add a third opinion: it is really not a good idea for you to publish an article against the author's wishes, even though you hold the copyright. If you did this, then at best you would never have any dealings with this author again. (In reality: that outcome must be acceptable and perhaps even desirable for you to contemplate this course of action.) More likely, the author will complain about it to a smaller or larger number of people, creating a bad name for your fledgling journal. For what it's worth, if I heard a story about this from an open access journal published by a regional scholarly society: I would not do business with you.
If I am honest, I didn't find your reasons for not publishing the change to be very convincing. It sounds like you are viewing this as a power struggle with the author that you want to win. You write:
He insists on adding new content and a new reference to the main text of the article, which I, as editor, will not allow.
Why not? You don't really say why you care so much.
He is offering to resubmit the manuscript with the new content included, and to wait for the results of a completely new peer review. He is very persistent about including the new content which, I think, has no impact at all on the overall quality or on the conclusions of the paper.
If in your professional opinion the added content has no impact on the quality of the paper, then in particular it has no negative impact. So what's the problem?
The offer to get a completely new peer review seems like a generous move on the author's behalf. He's saying that he's not trying to sneak anything by you but rather wants to err on the side of going through the process thoroughly, even to the extent of having the publication delayed (surely) or jeopardizing the acceptance of the paper (possibly). But because you feel the change is so minor, you can just bypass this entirely. It seems to be matter of retypesetting the paper.
The reference is a fresh (2015) self-reference, so I assume that he simply wants to increase the visibility of that other paper.
This seems like a rather bad faith assumption. If the author feels that his other
work is relevant, it contributes to the literature to have the citation appear. In general, it can be tricky to get cross references in one's own recent papers right, and last minute additions can be very helpful here: if it's a 2015 publication then it is plausible that the author didn't have the full bibliographic data until now.
It would be a waste of our journal’s resources to disregard that we have the copyright of a manuscript, which we accepted for publication after rigorous peer review.
That's written as if you're conveying information, but to me it sounds like a dogmatic assertion about you've already decided to do. What resources would be wasted by retypesetting the paper? Let me quote from another answer which approaches this issue more reasonably:
If the other last-minute changes would increase the typesetting and copyediting costs, then you could ask the author to cover the increased costs. (That may upset the author, but it's a traditional approach to handling this situation.) You might also keep costs down by adding the material as a "note added in proof" at the end of the paper.
In other words: if it is really a waste of your resources, ask the author to financially compensate this waste. Or tell him that you need to accommodate him in the most inexpensive possible way unless he is willing to financially contribute. Either of these is so much more reasonable than publishing a paper that the author doesn't want you to publish. You speak of being "strongarmed" but many authors would regard the behavior you're contemplating as an extreme example of that. Can you not work with the author to come up with an outcome in which neither is strongarming the other?
Added: I just read the bit about the added publication coming from a predatory Open Access journal. I think that probably should have been part of the question, since as my answer above indicated, the explanations by the OP didn't really explain the motivation. This does. However, I don't think it really changes my answer, except to say: either you have concern about the added content or you don't. If you do, you should get it re-refereed. If you don't, then the fact that the citation is to a not-so-great journal is really not your concern: citations should not be censored (which is not exactly what's happening here, but just to enunciate the principle) by the editor because they are unseemly to the journal.