As a white British person I have no experience of having to consider how my 'ethnicity' should be represented. Black, Asian, and Minority Ethnic - BAME - has been used to group people together. Of course, there are many strands to ethnicity and it's helpful to have a common and accepted phrase or word to describe a group.
In the US, 'people of color' is a commonly used term that is often seen as disrespectful in the UK, due to the historic use of the word color. 68% of people currently classified as BAME would be happy with 'Ethnic Minority' as an umbrella term, so maybe it's time to move to what the clear majority prefer.
BAME, like “black” and “coloured’ before it, was doomed to fail because it’s impossible to distill centuries of history and culture into a handy acronym.