I am sure you as a Muslim know that worship and prayer is the greatest and most effective tool to battle anxiety and stress. As the holy Quran says, with the remembrance of God hearts will become tranquil. By someone seeing prayer as a burden or something that would hinder their mental stability, or even sleeping pattern, this shows they have not yet discovered what true faith and worship is.
The best time to gain focus in life, or rizq, or mental and spiritual stability is fajr time. Adopting a healthy and religious sleeping pattern will enable one to get good sleep, and also fuflil their basic religious requirements.
Of course, it will mean that one would have to change certain things in their daily routine, like avoiding things that keep them awake at night, avoiding caffeine, sleeping early, creating better atmosphere at home, and so on.
There are many ways of getting your good sleep, and also performing fajr prayer in its proper time. This means that the problem is not fajr time disrupting the sleep, but rather having a better routine and stronger reliance in the Almighty.
With prayers for your success.