Why will they fight? No, seriously, why?
The idea of a Mihawk vs Shiryu battle ever happening pretty much relied on the idea that he would join Shanks in a battle against the BB crew. What? Will they now attack Buggy just so this headcanon of Shiyru ever amounting to more than a stepping stone works out?
Because Mihawk has NO devil fruit, Shiyru has never been said to be some sort of sword collector fanboy like Tashigi, and the sword he does use has been given a name by Oda as well be declared as one of the Meito so him just ditching it in favor of Yoru also seems unlikely.
So, again, why would this fight even fucking happen? Because the chance of the Black Beard pirates all of a sudden deciding to attack Buggy and his newly formed Yonko group, and in your scenario winning and killing Mihawk very shortly after the fucking group has been introduced is... well it's not even bold it's just weird. Oda is clearly dead set on continuing his fucking running gag of Buggy failing upwards so chances are? He, Mihawk and Crocodile will just manage to luck their way towards the One Piece where Zoro will have his duel with Mihawk.
That scenario does sure as fuck makes a lot more sense than the BB pirates starting shit with Buggy and wrecking his crew despite only being introduced as proper Yonkous literally last chapter and killing the one guy whom they have NO beef with that we know off, who DOESNT have devil fruit powers, all just so some guys can have their "duel to the death" with "tension" as if that shit exists in the first place with Zoros final duel.
He WILL win people. So do not come to me with "high stakes" to "tension"when it comes to the WSS title as if it matters, especially with regards to Zoro and Mihawk. He would have ALWAYS let Mihawk live, long prior to him being his master because Zoro never kills his opponents and he already got a favor to return in the sense that he was sparred.