Is the circulated story about the Companion who uttered three words before his death ("Would that it were new," "Would that it were far," "Would that it were complete"), which the Prophet, peace and blessings be upon him, interpreted as his good deeds, an authentic hadith?

The statement mentioned in the question is not a prophetic hadith, nor a saying of a Companion or a Tabi'i. It has not been found in the reliable books of scholars, and it is most likely a circulating falsehood. It is obligatory to beware of such fabrications and not to spread them, but rather to erase them and replace them with authentic hadiths that encourage charity. The Messenger of Allah, peace and blessings be upon him, said: "It is sufficient falsehood for a person to narrate everything he hears."