How should we deal with a brother who has abandoned prayer and has been advised multiple times without responding, and is it permissible to contact one of the shaykhs or students of knowledge to advise him?

This person is in great danger and exposed to God's punishment. Strive to advise him and clarify the danger of abandoning prayer, and that, according to some scholars, he is outside the fold of Muslims, and that he is worse than the adulterer, the thief, the wine-drinker, and the murderer. Mention to him the worldly and otherworldly benefits of prayer. Use appropriate means such as brochures. If he benefits, then praise be to God. Otherwise, guidance is in God's hands. If he is deterred by boycotting, then boycott him. Otherwise, do what is most likely to elicit a response from him. Communicating with students of knowledge is a good thing.