I don't know if you still care, but I think I found the answer to your question.
Look at Proposition 1 in Chapter 14 of Quantum Groups and Their Representations by Klimyk and Schmudgen. It shows that there is a bijection between left-covariant first-order differential calculi over a Hopf algebra $H$ and right ideals of the kernel of the counit of $H$, and it shows how the relations in a first-order calculus are obtained from the ideal. I have never worked with these things, so I don't know how tractable the computations are, but as far as I can tell from quickly scanning it seems that all the maps are at least explicitly defined.
As for higher order calculi, I am not sure whether/how these results extend. But at least maybe this is a good start?
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