You can't read boy.
BB didn't run from Ray, he was already asking a deal with Boa while talking with Koby to get his men back but couldn't trust her. With Rayleigh coming, he had his hands full with Ray+Hancock in 2 v 1, and Ray proposed a deal to give his men back, and he accepted, if he didn't accept he needed to fight Ray+Hancock, which your precious Mihawk can't even dream of because of wanting a ''peaceful life'' and accepts being an underling to Yonko.
Still, he didn't give a shit about S-Hawk and Ray, and kidnapped Koby after getting his men back.
Tai lost all of his logic due to BB hate, I debunked his nonsense too easy.