What is the ruling regarding the Fajr prayer performed with doubt about the onset of menstrual bleeding, given that menstrual bleeding certainly began after the Dhuhr prayer on the same day?

If the brownish discharge is mixed with blood during the time of menstruation, meaning there are 15 days or more between it and the previous purity, then it is the beginning of menstruation. Therefore, your Fajr prayer was during the menstrual period, when prayer is prohibited. We hope that you will not bear sin for your ignorance of this.