Do you know how the sharingan predicts movements and tracks movement?
The sharingan tracks speed and follows movement better than anything else, but it predicts movement based off of subtle muscle movements. The sharingan notices this and predicts where the opponent will be because of it.
But with puppets it wouldn't make any difference as they don't have muscles and so the sharingan can't predict their movements just like Sasuke couldn't predict the movements of Naruto's tailed beast chakra in his tailed beast cloak.
So Kakashi could see movement of Sasories hands and fingers, and predict, what will do the puppet. It is as at list. Sharingan help predict the movement, even if you take it out with puppets, Kakashi still see every move from them and can dodge. He is well trained and not limited to use doton.
As for Sasuke that time he had only 2 tomoe if I am not mistaken and gaining
3rd he managed to track Naruto's movements.
Overall it wouldn't be that hard to dodge all of them, but even if he gets scratched he will die soon as the poison will paralyze him.
Sakura and Chiyo were the perfect opponents for Sasori because Sakura had an antidote and Chiyo knew exactly how Sasori fought, being his grandmother and a puppet master herself. Kakashi would have neither of those advantages.
Yes he has no antidote and not puppet master, but he has very good trained body, sharingan, that can see chakra flow, and 3 elements (water, lightning, earth)
And what a staff of Kamui take only external part? Kamui send the object (or its part) seeing in its center to another dimension. So if Kakashi kamuis puppet with hart it will be demolished with every thing in sight. The only chance Sasory through hart at the last moment to another puppet. But this is also could be difficult and Kakashi will see it as well. So Sasori even if he wins not with ease and with cost of a lot of his puppets.
I am for Kakashi. More chances.