Chofetz Chaim seemed to avoid machlokes in many issues.
I went through 2 volumes of his letters and public speeches in 1920s Poland and he calls on people not to send kids to non-religious schools, namely cultural and such, and talks about various anti-religious movements in Poland, but he has nothing to say about Zionism as such.
Other sources say that he did not publish anything about Zionism until he saw articles saying that one can speak Hebrew without Torah and it is a maala, etc and Chofetz Chaim finally responding publicly saying that it is a mistake to think that EY without Torah will work. As to RZ, there seem to be multiple quotes of CH.Ch. welcoming R Kook’s son, of him refusing to sign up to denunciations of R Kook, they do not like they are made up.
I think, whatever the disagreements, we should at least take to heart Chofetz Chaim’s attitude of not loving machlokes.