What are the long-term impacts?
Cossack noblemen might have more success in blocking the Tatar slave raids, enabling a more rapid settlement of the Wild Fields by Ruthenian peasants rather than Muscovite ones, and the establishment of a string of fortresses in southern Ukraine. Ideally, for Poland, that would result in more Tatar slave raids against Muscovy. With the southern flank secure, the Commonwealth might have more success against Muscovy during this century, or against Sweden. Potentially, they could keep Muscovy landlocked perpetually.
I will not answer how, even if it is the most interresting part, but I need time to study the OTL before thinking about a good ATL.
But I imagine that between 1621 and 165X..., the Cossacks with or without the direct armed intervention of the Commonwealth armies managed to invade and conquer the Crimean Khanate, while staying subjects or rather good allies of the PLC.
So what after :
Did the Tatars slavery raids are stopped ?
I doubt because even if Crimea is conquered, the surviving Tatars can raided the PLC left bank (of Dniepr) from their territories north of Azov Sea, and from Ottoman territories of Jedisan, the Nogai Hord can raid Braclaw and Podolia, the Nogai can even penetrated in Red Ruthenia by following the Dniestr...
Next problem :
A cossack semi-autonomous region in Crimea and near the mouths of the Dniepr still represent a magnet for every ruthenian peasants fleeing the oppression of the servage... It also represent a major social headache for the PLC social order.
So next :
What prevent the hate and the hostilities between the Polish magnates in south-eastern Ruthenia and the Cossacks to engage in politicals or even military struggles. The Cossacks can still decide to attack the magnates (privates wars between PLC magnates was maybe uncommon but still possible) while still respecting their oath of alliance or loyalty to the King.
Next :
The Cossacks will be even more willing to attack by sea the coastal Ottoman territories so the Commonwealth will have a never ending series of diplomatic troubles or wars with the OE...
Next :
What prevent Russia to ally with the semi-autonomous Cossacks champion of the Orthodox Faith in the PLC in a major war against this country ? The Cossacks being able to gain a full independance and territories populated by orthodox ruthenians peasants for whom they are heroes.
And it is only the beginning of my brainstorming.