Would it be weird to tell my adopted brother that he's not part of our family after he demanded to inherit the family business?
Why would you tell your brother that?
He has no business sense and would likely ruin it.
I understand you want the business to keep succeeding, but who inherits your parents' business is not your decision to make. Be grateful for what you have.
He's just a kid, I should just tell him to piss off.
While being upset is understandable you could really hurt him emotionally by telling him he isn't even part of the family. He may be in the wrong for trying to get the family business but either way it's not your choice to make. Both of you are being rude to each other and your parents so try to be more considerate.