I think you could use considered, but only if you were able to change the order of the sentence.
If you have to fill in that specific gap, then only A is natural.
- He wasn’t asked to take on the chairmanship of the society, being considered insufficiently popular with all members.
- Considered insufficiently popular with all members, he wasn’t asked to take on the chairmanship of the society.
I'm not sure there's really a "rule" to explain the difference, but I would say that if you removed the "being" from option (1) but didn't rearrange the word order, it sounds as if the *chairmanship*, not *he*, was considered insufficiently popular, which of course makes no sense :) If you invert the order of the clauses (2), it's clearer that it refers to him, and so possible, although I still think the first version sounds more natural.