Certainly a person can be sublime. I disagree with those here who say a person can't be sublime, especially since you call him your brother, which is more than just a person but a relationship. Calling your brother "sublime" would mean your brother is awesome, the epitome of excellence, or beauty incarnate, possibly pointing to how he is as a brother, i.e., the perfect brother, the exemplar of brotherhood.
That said, I do have one issue. Since "sublime" has a connotation of being complete and absolute, even quintessential, it doesn't have degrees. Something or someone is either sublime or not sublime. There is no middle ground. It's kind of like being unique, which means one of a kind. One of a kind is one of a kind. You can't be more one of a kind than another one of a kind. There's only one or not only one. That's it. It's like a switch, on or off. So I'm a apprehensive about you saying your brother is "becoming" sublime. While I can't categorically say it's wrong or that there is no context where that wouldn't be right, it nonetheless smacks of wrongness. That's why I think so many people here have told you no, because it sounds wrong, even if they couldn't quite put their finger on the right reason why it sounds wrong.
You can properly say, "My brother is sublime," but I can't endorse you saying, "My brother is becoming sublime," any more than I could endorse you saying, "The light is becoming on," because it's on or it's off, because he's sublime or he's not. The quality of being "sublime" is static, not processive.