Which is better: performing the Fajr prayer in the mosque or at home to ensure it is performed within its prescribed time, given the doubts that it is being performed before its time based on astronomical calculations?

Prayer is not permissible unless the individual is certain, or strongly believes, that the time for it has begun. If there is doubt about the time having begun, the prayer should be delayed until one is certain or strongly believes that it has. If you doubt the start of Fajr prayer time after investigating, delay it until you are certain or strongly believe that the time has begun, even if this means performing it at home.