In my opinion, it would help if one of your recommendation letters addressed this issue in just the right way. Exactly what that means is a bit delicate, but let me take a try:
I wanted to give some context for the absence of a letter from Professor X in Dr. TMTTM's application. Professor X and Dr. TMTTM unfortunately did not get along. At a certain point the university intervened. No wrongdoing was found on the part of either party, and Dr. TMTTM was set up as a kind of independent researcher, a situation in which Dr. TMTTM flourished. I won't [better, can't] speak to the particulars of
the relationship between Professor X and Dr. TMTTM, but I can say that in my experience Dr. TMTTM has been quite collegial. Indeed...
Now of course you don't write your own recommendation letters, so this is not copy for you; rather I just mean to claim that having a bit of explanation could smooth things over for you. In my mind, of all the reasons for your supervisor not to write for you, the fact that you had a personal conflict after which you did all your work independently is really a good one. With no explanation, I might have imagined something worse.
I suppose the above advice is predicated on the idea that those who look at your application are going to see the lack of letter from your supervisor as a definite hole. If you get sufficiently good letters from other people, that may not be the case. Or you may just decide to try for an additional postdoc position, for which having a strong letter from your PhD advisor seems more natural and important. After you do one more postdoc, no one will care whether you have a letter from your last postdoctoral supervisor: you can only have so many letters.