Is there sin for one who refused to fulfill a need for which its owner invoked the Prophet (peace be upon him and his family), and then fulfilled it later through the intervention of the manager? And what is the expiation for that?

There is no sin upon you for what you mentioned. Even if he were to ask you by Allah and swear an oath upon you, and you did not do it, you would not be sinful, because fulfilling the oath of one who swears is commendable, not obligatory.