I don't think it's bad blood as much as it's a pirates devotion to their crew. Since they were potentially on opposing crews, they had to fight each other. And while it may have started out as disdain for a different crew, it did turn into a mutual respect as swordsmen. I mean, if Mihawk still had bad blood with Shanks I doubt he would have stayed and drank with him when he went to visit Shanks during the AP arc.
Also, to address the whole not fighting to the death thing, they might have. It is stated earlier on that Shanks was on Mihawks level as a swordsman when he had both arms, but that once he lost his good arm, Mihawk quit Dueling with Shanks. Now if Mihawk and Shanks had been dueling since they were both in their respective crews, and if Mihawk was in WB's crew, then that gives implication to the idea that both Mihawk and Shanks grew at the same rate when they were in their respective crews. So through time, while they were indeed fighting to the death every time GR and WB met up, they were always evenly matched and couldn't kill each other, even though they still tried. Then through time, as they made what seems like a sort of acquaintanceship or even a friendship, the bad blood was turned into respect as swordsmen, then when Shanks lost his arm, Mihawk couldn't kill Shanks because of his swordsman's honor and the respect he held for Shanks.
