Oda gave Mihawk weird personality because of "Swordsmen duel" agenda.
When we first meet him he's chasing a pirate fleet because he was bored but now he doesn't want to be bored by some random fodder marine fleets?
Oda needs to make up his mind.
Also, I wouldn't reach too much into him not wanting to chase Luffy, Shanks or BB.
He told Shanks he doesn't care about him since he's crippled. Luffy nor BB are swordsmen so he has no reason to fight them + he said becoming a Yonko is boring.
He already has to deal with people mistakenly hailing Buggy as a Yonko and them sailing on the same ship.