In my opinion, this practice is actively discouraging scientific exchange, and it seems simply excessive to ask for another 4000 USD.
It sounds like the journal treats comments exactly the same as standard articles and applies the same blanket fee. Unless they claim / purport / promise to do otherwise, they must have the right to do this -- how could they not? And correspondingly you have the right to regard the practice as dodgy / greedy / actively discouraging scientific exchange and not to pay to publish your reply to the comment. In my opinion you should seriously consider this.
Furthermore, should the reply not be considered part of the comment, therefore, if anyone, the commenter should pay for the publication cost?
C'mon, show a little empathy. If you had written the comment you could have said "should the comment not be considered part of the article, therefore, if anyone, the original authors should pay for the publication cost?" The two sentiments make the same amount of sense to me.
Bottom line: don't pay $2000 to reply to a comment if you feel that you and the academic community are not getting $2000 worth of value. There are lots of venues to publish comments on journal articles, ranging from websites that seek to collect such things to your own homepage. A comment or a reply is probably not going to be viewed as a real publication, it's probably not going to be peer reviewed in the same way (or at all?), and so forth, so just because you want to pay to play ball with this "renowned" journal does not imply that you want to do this too. If you feel strongly enough, push back more than you have. Write to the authors of the comment and offer to host both their comment and your reply elsewhere, free of charge. Copy this correspondence to the journal editors, and lament the unfortunate nature of the situation. (Don't do this lightly, and understand that you may mildly annoy some of the editors by doing this. But that is also your right...)
Perhaps I should end by saying that I work in an academic field, mathematics, for which (i) the notion of authors paying to make articles open access is just starting to become a thing and (ii) journals virtually never publish comments to others' articles. Just FYI.