How can I rewrite this sentence to avoid cliches and or biased language? My brother is a male nurse.
As in your last sentences, there is nothing wrong with the sentence, "My brother is a male nurse." However, there is no reason to specify that he is a male nurse. This is where the bias is. To say that he is a male nurse suggests that men should not be nurses or that the profession of nurse is a job left only for women. In light of this, here are two alternatives.
First, you can simply say: "My brother is a nurse." This is straightforward and to the point. Most likely this is the best alternative.
If you wanted to be general, then you can say: "My brother is in the medical profession." Or you can say: "My brother is the health industry." Both of these are a little more ambiguous in terms of what he does exactly, but that is something that you might want.
If you wanted to be more specific, you can say what kind of nurse he is. For example: "My brother is an pediatric nurse."