Is specifying a particular day for ghusl and repentance considered an innovation (bid‘ah), and must a person afflicted with obsessive doubts (waswas) ignore these thoughts and take the initiative to repent and perform ghusl to rid himself of the ghusl-related waswas that delays his repentance?

You must repent sincerely and immediately to Allah Almighty, and do not pay attention to the whispers of Satan that repentance is not accepted except on a specific day, for Allah accepts repentance at all times. You are on the religion of Islam and have not departed from it, so there is no need to perform ghusl (definition) to enter Islam. Do not pay attention to the whispers of Satan that you have committed one of the nullifiers of Islam, but rather repent to Allah. Turn away from whispers in any aspect of religion (creed or acts of worship) because they are from the cunning of Satan, and remain steadfast in Allah's law and strive in supplication so that Allah may remove this illness from you.