Is repeating the verses of Surat Al-Falaq and the recurring supplications in the morning and evening remembrances in this manner Islamically permissible?

Presence of heart and contemplation are essential when reciting adhkar (remembrances) to mend the heart. It is permissible to repeat a dhikr to attain khushu' (devotional humility), so long as it does not lead to waswasah (obsessive whispering). Ibn 'Uthaymeen affirmed the permissibility of repeating Al-Fatiha in prayer for this purpose.

As for reciting the Mu'awwidhat (Surahs Al-Falaq and An-Nas), it is permissible to recite them together and then repeat them, or to recite each Surah three times. However, repeating a single verse three times until the end of the Surah has not been narrated from the scholars. It is therefore best to avoid it, unless the goal is to attain khushu', understanding, and contemplation, in which case there is no harm, as long as it does not lead to waswasah.