AAQ:
The gedolim of his time, who needed his services as “Chief Rabbi”, obviously had to address him as he expected.
The link describing the Brisker Rav as mentioned above, if the mods allow the link, is: https://yoel-ab.com/katava.asp?id=130
That link also mentions that Rav Chaim Brisker, even back in Europe, did not believe that Rabbi Kook fasted the many fasts that he was reputed to have fasted. It also notes that the source (which Chaim here loves to quote), about how the CC protested against those against Rabbi Kook, is none other than Rabbi Kook’s son Rabbi TY Kook.
The gedolim who did not need Rabbi Kook’s services were vehemently against him and referred to him as a “Navi”, like:
Rav Yosef Yedid in “???? ???????? ????? ????? ????”.
If the mods allow this link: https://hebrewbooks.org/pdfpager.aspx?req=22671&st=&pgnum=441
It’s a long piece, and worth reading.
The Satmar Rav labeled him with all sorts of names like “michabel biKerem Hashem” and much, much more. Both the Satmar Rav and Rav Elchonon Wasserman (in Kovetz Maamarim) labeled Rabbi Kook a “rasha gamur”, based on two different pieces (one each) from Rabbeinu Yonah.