Kağan is the correct spelling but we can think of the word as a stylized version of it used for practical purposes while actually Turkish language does not use double vowels like aa. I like it.
There are things which is not only related to logic.
Kağan is correct for Turkiye language,but there is issue. This fighter will be part of other nations. This fighter name have to be suitable to be pronounce in english. This fighter name have to inspire other nations All Turks, All friendly by history, All allied also nations which hate Turks. There for all know word "Kaan".
Conclusion is that Kaan is Turk word and all other nations know this word.