What is the ruling on consistently reciting a specific dhikr in rukoo' (bowing) and sujood (prostration) that was not reported from the Messenger of Allah, peace and blessings be upon him? Is this action permissible or an innovation?

There is no harm in reciting adhkar (remembrances of Allah) during bowing (rukūʿ) and prostration (sujūd). However, one should not commit to a specific dhikr that has not been designated in the Sunnah for these positions, lest it becomes an innovated addition (bidʿah iḍāfiyyah).

The adhkar that the Sunnah indicates should be committed to are: "Subḥāna Rabbī al-ʿAẓīm" (Glory be to my Lord, the Most Great) in bowing, and "Subḥāna Rabbī al-Aʿlā" (Glory be to my Lord, the Most High) in prostration (three times is the minimum for perfection).

It is recommended to make abundant supplication (duʿāʾ) during prostration, both with those narrations that are transmitted (maʾthūr) and with others, provided they do not involve aggression or sin, based on the Prophet's (peace be upon him) saying: "The closest a servant is to his Lord is while he is prostrating, so increase in supplication."

As for bowing, what is required therein is the glorification and exaltation of Allah, based on the Prophet's (peace be upon him) saying: "As for bowing, glorify the Lord therein."

Some jurists have explicitly stated the dislike of committing to a specific supplication in prayer unless its meaning is general.