BAME is one of those words we often feel very mixed about. 'Non-white' is problematic (centres whiteness). 'POC' is often deemed too American (and 'coloured' has baggage). BAME is useful in *some* contexts but it's also imprecise.
I've heard 'global majority', but it denies the experience of being a minority in many cultures and places around the world. But also, Black people aren't a minority in, say, Ghana. The terms have to be specific to the place they refer to.
Someone suggested 'people with melanin' as a semi-serious option. That made me think of 'MelaNation' as a pun for some kind of project further down the line.
Language has also changed. 'Black' itself still means different things to different people. The solidarity around 'political Blackness' was, and perhaps still is, very important for decades in the UK.
I'm not sure what the answer is. Sometimes we need shorthands. Sometimes me need to refer to black and brown people as a collective with shared experiences of race. BAME is imperfect for that.
For the moment, then, it seems like any term is a compromise. That will likely continue until we can find more precise and nuanced language. I personally kinda like POC, but I know many hate it almost as much as BAME (and occasionally moreso).