Must I re-disburse the Zakat I paid to the mosque, knowing that I was unaware of its prohibition?

It is not permissible to disburse Zakat (definition) except to the eight categories mentioned in the verse from Surat At-Tawbah (definition): "Zakah expenditures are only for the poor and for the needy and for those employed to collect [zakah] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allah and for the [stranded] traveler - an obligation [imposed] by Allah. And Allah is Knowing and Wise." Zakat must be re-issued if it was disbursed to someone not entitled to it, and what was disbursed to the mosque is considered an act of charity.