They aren't, wrong assumption here
Also there won't be a war between Buggy and Shanks, please read One Piece again and how the Shanks-Buggy dynamic is written in the series
Here's my headcanon why Mihawk joined Buggy: Mihawk's only objective is to be left alone, he retreated to an island with only Baboons to be left alone, and that's why he wanted the Shichibukai title. So, by banding together with Buggy, the WG sees their group as a bigger threat than individually (I assume former Shichibukai like Crocodile also banded together under the "Buggy" label"). Now, Buggy is a Yonko and Mihawk reached his goal, other pirates fear the Yonko and will leave Buggy+allies alone, and the WG is busy with other stuff and will most likely leave them alone as well, at least the lower ranks of the marine below vice-admiral. And that's it, that's all there is to it, just Mihawk wanting to have his peaceful and quiet days again.
I still have no idea where you get the idea from that Mihawk is stronger than Shanks, there is literally 0 evidence for that in the actual story, and 100% evidence for it all in your head.
(also fuck Powerscalers, you guys are so annoying and most of you are literally 15, fuck off)
