I have never heard this Lvush story before, so what I am just saying here is conjecture.
Sometimes stories are told over and people dont convey the facts correctly.
There was no IDF in Rav Kook’s time, but there was paramilitary groups and paramilitary uniforms. He could have been wearing a Paramilitary uniform of a precursor to the IDF and when the story has been repeated, someone mistook the paramilitary uniform of a pre-cursor to the IDF as and actual IDF uniform