Fujitaro would win.
Just because Mihawk is the World's Strongest Swordsman, doesn't mean he is stronger than Fujitaro. Fujitaro is an admiral with a very hax DF ability, who just so happens to carry a sword. Thats like saying Yassop (the worlds greatest sniper) could beat anyone in the OP verse who just so happens to carry a pistol on their hip. All Mihawk has is his swordsmanship meanwhile Fuji has swordsmanship, DF ability, and Admiral level haki.
There is a certain level of overrating of Mihawk, especially those who believe he is yonko level because he dueled with shanks. Please understand that dueling with someone is not the same a fighting all out against someone. Shanks would wreak Mihawk if it was an actual battle to the death.