Is it permissible to work as a partner in a shop that sells both permissible (halal) and prohibited (haram) products (such as alcohol), knowing that the partner's section will only sell halal products, but the profits are mixed and then distributed?

Your brother is not permitted to partner with these two individuals, because that would involve participating with them in selling alcohol and condoning an abomination. It makes no difference whether he directly sells alcohol himself or not, so long as he agrees to be their partner in this establishment and to share in the profits, regardless of their source.