Is a menstruating woman who has pledged to God to perform a specific dhikr after every prayer obligated to perform the dhikr after the obligatory prayer times, or can she perform it at any time of the day to complete the count, or should she make it up after her period ends, or is she exempt from making it up?

A covenant with Allah to perform an act of obedience is both a vow (nadhr) and an oath (yameen), and it is obligatory to fulfill it and beware of breaking it. Oaths and vows are bound by intention. If your intention was not to perform the dhikr except after prayer, or only on days when you pray, then nothing is binding upon you during menstruation. However, if there was no such intention, then what matters is the underlying reason, which is that you intended to perform a specific number of dhikr repetitions daily. In this case, it is binding upon you to perform that during menstruation, at any time of the day.