It would suuuuuuuuuuck, but if you really want to grab a Mihawk (you do), then you could use your Vivi and a friend's Laboon to grind for one. It's going to take you a while to finish him, but it's probably doable.
All you need to do is have your total HP be lower than 2x your Vivi's recovery rate as captain. That's going to be pretty rough for you, since your Vivi isn't evolved all the way, but it's still doable. The other option would be to try to farm a Laboon today, but that might be pretty hit and miss too.
Again, it's not going to be easy to do, but if you really want a Mihawk, then I do think it's at least somewhat doable.