Is it considered that assisting a refugee organization in paying for a pair of eyeglasses from ṣadaqah funds is forbidden for the Household of the Prophet (Ahl al-Bayt), and is this pair of eyeglasses generally considered ṣadaqah, and do genealogy books (kutub al-ansāb) affirm the lineage of a tribe to the Household of the Prophet?

Charity (sadaqah (definition)) is giving money seeking reward from Allah the Almighty. If the entity providing the eyeglasses intends thereby Allah's reward in the Hereafter, then it is considered charity, regardless of whether you are from the Ahl al-Bayt (the Prophet's Household) or not, and regardless of whether you accept it or not. However, if they do not intend the reward of the Hereafter, then it is not considered charity. The majority of scholars hold that the Ahl al-Bayt are not forbidden from accepting voluntary charity. As for relying on genealogy books to prove lineage to the Ahl al-Bayt, they can be relied upon if their author is trustworthy and has based their work on legitimate proofs of lineage.