Does the discharge of thread-like remnants of menstruation after performing ghusl and cleaning with a vaginal douchette after the Fajr Adhan mean that the fast is invalid, or is the fast valid?

This blood that has come out is undoubtedly menstruation. As long as it appeared after dawn, your fast for this day is invalid. You must wait for purity by dryness. You are required to make up this day, because the presence of menstrual blood during any part of the day prevents the fast from being valid.