There are three separate issues here that I can see, and two of them suggest a problem with peer review in your case.
First, most (all?) reputable journals and conferences have policies requiring confidential peer review. That is- reviewers are not allowed to share the work or discuss the work with until it is published. This gives the author the opportunity to present their best possible work to the community, and prevents low-quality work from entering the academic discourse without the validation of peer review. If your work was shared outside of the peer review system, it indicates a major breach of protocol.
Second, most (all?) reputable journals also have policies about conflict of interest in peer review. At a minimum, all such conflicts need to be declared to the editors involved, and ideally anyone with a conflict should not be a reviewer. This is not always possible, such as when the topic concerns a very narrow specialty, but at a minimum it is a sensitive situation that the editors should have known about.
In order for this person to have gotten your work, it would seem that one of these two best practices was not adhered to. If your field is actually incredibly narrow in scope then perhaps the review was allowed to be conducted by the person you were critical of. It is certainly worth a letter to the editor and they certainly should not be surprised by it.
The third issue is the timing. This is not a problem, except as it implies a breach of protocol in points one or two above.
Once the work enters the public discourse then it is free game for comment. The person you are critical of did the right thing by delaying their publication until your work was published, and that is all you can really ask of anyone. If we are to have confidential peer review then there must be a point at which the restriction of confidentiality drops (in part because the reviewers are often the best qualified people to participate in the public discussion that follows the private review). You can see this especially with high-profile papers when they are published- all of the people who you think would normally be commenting are silent, then suddenly the paper is published and they all give a big sigh of relief and say, "OK, now I can talk about this."
As an author you might understandably want your work to have a "grace period" where it can be discussed freely without rebuttal, but as far as I know that's never been a thing. If you have to pick a point in time to drop confidentiality, then the public publication seems to be the best.