What is the ruling on slaughtering the "fidyah" on the last day of Ramadan, and does it have a basis in Islamic law or is it an innovation?

The fundamental principle concerning acts of worship is to pause and restrict oneself to what the Lawgiver has prescribed. Therefore, if sacrificing is specifically designated for the end of Ramadan, based on the belief of its superiority or proximity [to God], then it is an innovation (bid'ah). This is because no text supporting it has been reported in the Book of Allah or in the Sunnah of His Messenger (peace and blessings be upon him), nor did any of the Salaf (pious predecessors) practice it. The Prophet (peace and blessings be upon him) said: "Whoever introduces into this matter of ours that which is not from it, then it is rejected."