As I mentioned in response to your previous question about ribbon elements, the element u which is defined from the R-matrix, u=\mu\circ(S\ot \id)(R21) has the property that uS(u)=v^2 (well this is not the formula I gave for u in that post, because the one I gave was incorrect; this one appears to be correct according to wikipedia).
This relation v^2=uS(u) is true in any ribbon Hopf algebra, and in particular it implies that v has to be a square root of uS(u). So I think this means that the ribbon element is almost unique.
More precisely, let v and w be two ribbon elements. Then v/w is a grouplike element of order two. I think this implies that the corresponding invariant applied to a link will be multiplied by the constant v/w applied to each link. Now if you choose irreducible representations to label your link, then this number would have to be +/- 1.
Does this seem correct?