Is continuing to recite "Subhanallah wa bihamdihi subhanallah al-azim, Astaghfirullah" one hundred times between Fajr and Subh, based on what the questioner read in the book Ihya Ulum al-Din, considered an innovation (bid'ah) or is it impermissible to continue doing so?

The hadith (definition) you asked about concerning the virtue of theing "Subhan Allah wa bihamdihi, Subhan Allah Al-Azim, Astaghfirullah" (Glory be to Allah and with His praise, Glory be to Allah the Most Great, I seek forgiveness from Allah) a hundred times between Fajr and Salatul-Subh (the dawn prayer) is not authentic, even though dhikr itself is great and its virtue is established by authentic hadiths.

Al-Iraqi, in his documentation, indicated that the hadith was narrated by Al-Mustaghfiri in "Ad-Da'awat" (Supplications). He mentioned that it has a supporting narration from Abdullah ibn Amr in Ahmad's Musnad with the wording: "And 'Subhan Allah wa bihamdihi' (Glory be to Allah and with His praise), for it is the prayer of all things, and by it, creation is provided for." Its chain of narration is authentic, but without being restricted to the specific number and time mentioned.

Therefore, one should not adhere to this dhikr with the specific number and time mentioned in the question, so as not to fall into bid'ah idafiyyah (innovating by adding details to an established act of worship). It is preferable to generally increase in tasbih (glorification) and istighfar (seeking forgiveness) and to rely on the authentic and established adhkar found in books such as "Al-Adhkar" by An-Nawawi and "Al-Kalim At-Tayyib (definition)" by Ibn Taymiyyah.