What to do when publishing a paper about mistakes in previous literature?
To begin with, I find it highly problematic that this journal has a policy of inviting authors to review any paper that criticizes their work. That seems to me to be highly inhibitory of criticism and likely to produce situations exactly like the one that you are facing. 

I find it doubly problematic that this author has chosen to communicate with you outside of the review process in order to persuade you to change your writing. His communications can easily be interpreted as an unethical quid pro quo request: you drop your criticism of his paper, and he'll drop his criticism of yours.

What is not clear from what you have written, however, is just what the journal intends to do with the review that they have received. Surely the journal must understand the potential conflict of interest here.  So is the review mainly to allow an author to point out errors in the critique that those less familiar might overlook, or are they going to treat it just like one of the other reviews?

I would thus recommend taking the following approach:


Write to the editor and let them know that the author has approached you separately with what you are concerned may be an unethical proposal --- the editor needs to know about this and should be able to make their approach clear to you. Ask the editor how they would like to proceed.
In parallel, use all of the constructive parts of the criticism to improve your manuscript. 
Consider slightly softening the language in your criticism in the manuscript. I do not know how you have written it, but it's often possible to use slightly less definitive language to make the same point, e.g., changing "Papers [1] and [2] should be re-analyzed to see if their results are affected." into "As it is possible that results might be affected, we would recommend considering re-analysis of papers [1] and [2]."  Same substance, but less confrontational and allowing you to write that you have adjusted the paper in response to the review.


Hopefully, the editor can provide you some clarity as to how to navigate the difficult situation that their journal has created for you, and judicious adjustment can allow you to pass peer review without compromising your ethics. If not, there are other good journals out there that will not have this unusual conflict-generating policy.