Is it permissible to delay the Fajr prayer beyond its prescribed time due to suffering from continuous tension headaches that cause poor concentration and lack of sleep as a result of taking prescribed medications?

The fundamental principle is the obligation to observe prayers at their prescribed times. Many Hanbali jurists have permitted combining two prayers for the sick. Therefore, you may combine Maghrib and Isha prayers by performing them early (Jama’ Taqdeem), then sleep and wake up for Fajr prayer before its deadline. Taking measures to wake up for prayer is a matter of differing opinions; the inclination is towards its obligation, but for necessity, what is disputed as forbidden becomes permissible.