Can you add a reference for your bound? I presume that bound is for the zero-error SVD? Are you only interested in that case? Do you care about the bit-size of the entries and condition numbers?

I am not an expert on this topic but in Dave Penneys' answer to this other thread it is mentioned that it is not known whether rank 4 fusion rings are categorifiable.

Good point. Yes, I agree with you now. Just for completeness, it does seem hard to classify conjugacy class hypergroups as well because classifying arbitrary abelian hypergroups / fusion algebras is hard (as you point out) and, on top of that, classifying finite groups is also hard (you can always determine the structure constants $N_{ij}^{k}$ of these hypergroups / fusion algebras if you know everything about the finite group conjugacy classes, but then you know a lot of information about the group already...).

Still, the conjugacy class hypergroups / unital based rings arising from finite groups, despite being already very broad objects seem to be simpler objects than arbitrary unital based rings. I'd like to know, do these obstacles you mention also imply (maybe suggest?) that conjugacy class hypergroups are also difficult to classify? That part was not clear to me (yet I need time to digest your answer).

I am interested on your answer. One first thing I see is that your claims are relevant to at least the first half of my question because the conjugacy class hypergroup of any finite group $G$ is a special type of fusion algebra / unital based ring normalized by the ring homomorphism $d$ you define (this is explained eg [here]). Fusion algebras can always to be renormalized to be abelian hypergroups, and you are saying these are already hard to classify.