How can one atone for a false accusation of theft and reclaim money from someone who was wronged, when that person cannot be found to have their rights restored and to be apologized to?

There is no sin upon you for accusing the man of theft, due to the strong circumstantial evidence: namely, his being caught holding your daughter's hand and the disappearance of the bracelet. However, after the bracelet was found, you must restore the man's right to him if possible; otherwise, give charity on his behalf. As for your donation of the money to the poor, it is not sufficient unless you intended the charity to be on his behalf. Ibn Taymiyyah mentioned that if one despairs of knowing the owners of usurped wealth or deposits, then one must give it as charity on their behalf or spend it on the welfare of Muslims.