Is it permissible to fast to make up for missed Ramadan fasts after forgetting to make the intention overnight, if only four days remain until Ramadan?

Fasting is not valid without an intention. The valid time for the intention is at night (from after sunset until just before dawn). If you did not intend to fast on that night, your fast is not valid, and a general prior intention to make up for the days after purification from menstruation is not sufficient for you.