Is it an innovation to designate a specific number of times for seeking forgiveness (Istighfar) and sending blessings upon the Prophet (Salah an-Nabi) — may Allah bless him and grant him peace — as a punishment for sins or shortcomings, and if so, what is the alternative?

Binding oneself to acts of obedience as a punishment for shortcomings is not an innovation (bid'ah). The predecessors (Salaf) themselves reported doing this; for instance, Abdullah ibn Wahb would oblige himself to fast or give charity as a punishment for backbiting. It is advisable to remember Allah Almighty frequently in all circumstances, and not to confine it to a specific act of disobedience, while also learning about the virtue of seeking forgiveness (istighfar) and the importance of sending blessings upon the Prophet (peace be upon him).