For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($\epsilon)$ corresponding to this calculus is $$ < a+ q^2d - (1+q^2),b^2,c^2,bc,(a-1)b,(d-1)c>. $$ This calculus can be shown to generalise the classical calculus on $SU(2)$ when $q=1$. Does anyone know of a (good) calculus (and its ideal) for quantum $SU(3)$?
I do not know whether it fits all of your requirements, but at least going by the abstract, some version of Woronowicz' result was generalized to all of the quantum groups of classical type in Differential calculus on quantized simple Lie groups, by Branislav Jurco.