What should be done regarding a family name that resembles the names of the Prophet's household (Ahl al-Bayt) in the questioner's country, where people consider him to be from them despite the absence of a clear lineage tree? And should the questioner claim this lineage or deny it, and which of the two is safer for him from sin?

It is sufficient for you to belong to the lineage to which your family and father belong. It is sufficient for this to be well-known and widespread. If your family belongs to the Household of the Prophet (Ahl al-Bayt) and this is well-known and widespread, then this is sufficient. You are not required to prove this with a family tree, nor to deny it due to not possessing one. It has been established that people are to be believed regarding their lineages.