Is the daughter considered an illegitimate child if the first marriage contract was conducted in a US municipality out of ignorance, and the second contract was later conducted in a mosque, and is expiation required in this case?

The validity of marriage hinges on the fulfillment of its pillars and conditions, not on the place where the contract is concluded. If a husband enters into marriage with his wife, believing the contract to be valid, then the daughter born from this marriage is attributed to him by lineage and is not considered a child of fornication, even if the contract was فاسد (irregular or invalid), by agreement of the Muslims.