How can one ascertain whether my brother is afflicted by sorcery, especially since he refuses any help and believes that what he is experiencing is a trial from Allah?

What your brother is going through is a trial from Allah. His realization of this is good, but it is not correct for him to think that he must wait until Allah removes it from him, for belief in divine decree does not contradict taking means. The Prophet, peace and blessings be upon him, said: "Allah has not sent down a disease except that He has also sent down its cure." Narrated by Al-Bukhari (5678). Therefore, you must convince your brother to seek treatment. It is advisable for him to pursue two paths: conventional medical treatment and investigating the matter of his sorcery by presenting himself to a legitimate ruqyah (definition) practitioner.