A couple of suggestions. Both variants are correct in isolation but the sentence structure as a whole is not clear.
-"playing with something in the other hand" means that he is playing with something that is in his other hand (but which hand is doing the playing is not specified, although people would assume his other hand).
-"playing with something with the other hand" means that he is playing with something using his other hand.
They are sort of saying similar thing, the difference is whether you want to emphasise what he has in each hand or what the hands are doing.
But the sentence doesn't flow well as it is. "something in hand" means he is holding something but does not necessarily mean that it's one-handed. As the sentence moves on to say what each hand is doing, it's better to say "X in one hand, ...".
"a large iced coffee mug in one hand" means he is holding a mug meant for iced coffee (or that it is a coffee mug which has been iced). If you mean to say he is holding a drink, you would say "a large mug of iced coffee."
Formally you would say "doing X like Y" but if this is literary or casual writing, stylistically it sounds more natural to say "Doing X, Y style". Finally, in "one of the hair clips" the word should be plural.
So as alternatives:
Ben is slumped in his chair, pasha-style, a large mug of iced coffee in one hand, playing with one of Lili's hair clip with the other.
Ben is slumped in his chair, pasha-style, a large mug of iced coffee in one hand, the other playing with one of Lili's hair clips.