bhe: I have asked many, many people who were around in his lifetime. His Talmidim almost all called him “The Rav” (R’ Schachter still famously refers to him as “Rabbi Soloveitchik”, but he’s a minority). His contemporary Rabbonim and Gedolim referred to him as Rav Yoshe Ber. His Seforim were published as Rav Yosef Dov HaLeivi.
His academic philosphical works were published as Rabbi Dr. JB Soloveitchik, because the academic convention of the time was to use initials like that. No one who learned from him dared call him that, as he was vociferously against using the same convention for Roshei Yeshivah (someone once related to me that he went on a 15-minute rant about Kavod HaTorah after hearing someone refer to the Sridei Eish as “Rav YY Weinberg”).
At some point, people decided that a great way to malign him would be to claim that he wanted to be referred to as “JB” because he cared about academic convention more than the Torah aspect of his life. It is a gross insult and blatant falsehood, though I guess when HaKatan insinuates that he was an Apikores and Machti Es HaRabim (because he says he changed the Mesorah to one of religion and Avodah Zarah mixed together), I really shouldn’t complain about the comparatively smaller insult of using a derogatory nickname.
(Edit)