I will give the perspective of someone who has been an independent reviewer brought in on a disputed manuscript.
You do not know what the reviewer wrote to the editor. In a recent manuscript where I was asked to review in the situation you are in, I provided constructive and actionable feedback to an author. I discussed where the author had adequately answered the reviewers concerns and had not met those concerns. And then in the private comments to the editor, I expressed my reservation about the manuscript.
Do not take the rejection personally. This is a normal part of academia. I had a paper rejected after three rounds of edits recently! A year in the review process and the editor decided enough was enough and cut our paper loose. We looked at the feedback from reviewers objectively and prepared the manuscript for a different journal.
My advice, there is little chance that you will sway the editor. Accept their decision and move on. Fields are small and memories are long. Take the reviewer feedback that you were given and feel good that you got solid feedback on a manuscript draft from scholars in your field. Then incorporate that feedback and resubmit to a different journal.
A word of warning about resubmission. Address the feedback of the reviewers before you resubmit.