Ascribing BAME status can be very subjective. Just as discrimination is imposed upon people because of others' prejudices, BAME status can be assumed by them because of the imposers' prejudices.
I'm Jewish. I have been told that this makes me BAME. It's a minority view, but both Jews and non-Jews have told me this.
My people may have come from the Middle East many generations ago, and again more recently, but I'm not convinced that makes me BAME.
My people have certainly suffered severe discrimination, and in many cases still do, but it's not the same. Accepting the label BAME feels appropriative of someone else's struggle. I would rather stand side by side with people than merge us all into one falsely equivalent group.