Is it permissible to make up the Fajr prayer before going to work at 6:00 AM, knowing that the prayer time is 4:00 AM, and waking up early for it leads to inability to sleep again, and consequently feeling extreme drowsiness while driving, and fear of causing an accident?

It is not permissible to intentionally delay the prayer beyond its prescribed time due to drowsiness or going to work. One must observe prayers at their proper times with all their conditions, as Allah Almighty says: "Maintain with care the [obligatory] prayers and [in particular] the middle prayer and stand for Allah, devoutly obedient." And whoever fears Allah - He will make for him a way out and provide for him from where he does not expect, and will make his matter easy for him.