Is it permissible for me to pray in the commercial establishment where I work, knowing that I cannot wake up for Fajr prayer due to extreme fatigue, and that my father does not trust the workers he employs?

It is not permissible to miss Fajr prayer by staying up late; rather, one must adjust work schedules or take measures to wake up. Congregational prayer is valid at one's location, and it is not obligatory to go to the mosque, but it is better and more preferable to pray in the mosque due to its immense virtue.