Remember when people theorized zoro's EoS fight will he someone with a diamond fruit?
Then haki got introduced
I don't think beating mihawk alone will grant him the title. He needs needs more fame for that.
An admiral, probably green bull, will be on his list before he even fights mihawk. Green bull represents the exact ideology zoro hates most as we've seen in his fight against king.
Shiryu, in an EoS fight, might make him cut 'nothing' but that's not an actual feat that would help him on other occasions.
My personal theory, bro might cut the moon. Many people would see the effect, that grants fame. During the events he beats Shiryu too but that won't be the actual fight that grants him his title.
And it doesn't matter if he fights mihawk before or after that. The awakening of Shiryu's fruit isn't something you come across every day - it's a very special singled out event that he or any other swordsman will never ever have again.