So far shanks' big moves have been attempts at deescalating conflicts... So far we have:
Shanks attemted to convince WB of calling back ace to avoid a confrontation with BB.
Shanks stopped kaido from making it to marineford.
Shanks actually stopped the war at marineford.
So... I don't think it makes too much sense that kaido's defeat at the hands of luffy would result in him pursuing BB. Nothing in the series even evidences shanks having an interest in taking out BB. Shanks is certainly no fan but the only reason I could foresee for shanks to take a stab at BB would be that he wants to stop BB from doing something absurdly reckless that would throw the world into chaos. Now, kaido's defeat could an will result result in the world's big players making big moves to which shanks could respond to. But there's a pretty big difference between shanks seeking out a conflict himself and shanks simply trying to deescalate a situation as usual.