Thinking about it, it may very well be within Mihawk's character. Dude already achieved his goal and spent his time drinking wine at the castle or hunting down pirates that disturbed his nap.
Now that the Marines are hunting him again, what is he supposed to do? He already said he trembles from excitement being the one on the run again. Plus being a lone wolf with no territories or crew. I guess from his perspective, flipping the table and turning from hunted to hunter as part of an organisation that rewards him for doing so is the best case scenario. If I was a powerful lone wolf chased by the marines, I'd absolutely go for it. After all, Mihawk has no mandate either. That is also evident given Crocodile is part of it despite saying he wouldn't serve under someone else, having his own agenda.