I think you are maybe attributing too much intent to these white middle class mums.
Speaking as a white middle class mum, who knows a lot of white middle class mums (and is is most likely to be mums who are registering their girls in ballet, let's face it) I expect their approach is:
"Oh that's a good ballet school and it is nearby. I'm going to put little Emily's name down. But these lists are always long. I'm going to put her name down at [alternative ballet school] too. Oh and also this gym class here... and this tap class here... oooh a drama group, yessssss!!!!...and maybe also piano. It's a bit early, but an instrument is desirable. And Brownies.... oh no, silly me, I already signed her up for Brownies at birth."
I think there definitely is some blindness about the need to allow BAME communities to carve out their own spaces, because probably most white middle class families WANT to mix with BAME families, and simply don't see it as a problem to sign up to a list. Which, as you say, is permitted by the ballet school.
I remember being accused of racism by a BAME colleague at a new job when I was 21, because I could not name "enough" BAME people who were my friends, to satisfy her. She said, my thinly diverse social circle demonstrated I was racist. I took it to heart. I had grown up in a white community where I had not actually met many BAME people (my second best friend at primary school was Asian, but at secondary there was not a single BAME student in my class). When i began work in central London of course I encountered a more diverse community and found more BAME friends naturally, but it struck me - should my mum have sought out a more diverse group of friends for me outside school? Our neighbours were Asian (the only Asians in the street back then) and we were friends, attended each others family weddings, shared meals, usual neighbourly stuff. But in the back of my mind, this social pressure to integrate has been there throughout my life, not just white people trying to centre themselves but if we ALL actively seek out opportunities to meet BAME people, because we are the majority, it's gonna look like an appropriation, an invasion, a white-washing of legitimate BAME opportunities.
Another example. A friend of mine had been unable to conceive so went through the hoops to adopt. One form required her to draw a diagram showing her social circles - from immediate family, to close friends, moving out to more distant acquaintance. The entire purpose of the diagram was to map the proportion of her social contacts who were of a different ethnic background to her. She was terrified she might "fail" the diversity test as it wasnt clear what the expected outcome was, but the intent was clear: please demonstrate you are not a racist by stating how diverse your social life is.
I'm absolutely NOT saying that these BAME clubs aren't being flooded by white people, but I do not think it is done with malign intent. There have been mixed messages for decades from society about these issues.
Now let's address ballet. It is a hot spot because it is a VERY well known issue in particular in ballet, that certain physiques and looks are preferred. One issue is, you will never see an overweight prima ballerina. Another is, you will never see a very muscular female ballerina. You never see disabled ballerinas. And so on. Similarly it has been rare to see black ballerinas.
You are comparing BAME with groups for LGBTQ or disabilities. Honestly I have never seen a group for disabled ballet dancers advertised, nor one for LGBTQ ballet dancers. And I've never really thought about this before. And I agree, I wouldn't send my DD to a ballet group for disabled or LGBTQ dancers. In the first instance I would assume she would be out of place at a club specifically for disabled children. And I wouldn't send her to an LGBTQ dance club because I would assume the club has some special interest focus not relevant to my DD (well, not yet, she is only 10 so who knows what the future holds).
I am starting to agree we need carved out spaces so children from different backgrounds and with different minority challenges can flourish. I wouldn't disagree with that. But I do think it is important to raise awareness without simply bashing white people in the way you are doing. It isnt helpful because you are creating more tension as white people will say, "look I just signed up on a list, I wasn't trying to offend or intimidate or crowd anyone, no one told me I shouldn't sign up, I just want my child to enjoy some activities in a racially mixed context because there arent many white people in our neighbourhood."
Does any of this make sense? Very hard to explain.