What is the legitimate Islamic solution for dealing with accusations against one's brother of sodomy and assault on honor, and is it permissible to refer the matter to a Sharia judge in a country that does not rule by Islamic Sharia, such as "Egypt"?

Advise your brother and call him to Allah, and explain to him the seriousness of what he is doing. If he insists and you are harmed by associating with him, then abandon him. You can also intercede through someone who has influence over him to deter him. And if the only way to avert his evil is to take the matter to court, then there is no blame upon you due to necessity.