Very concerning that there are people who pretend (1) that racism is no longer an issue in the UK when anecdotal evidence plus actual evidence shows that it is just as bad as even except it is now more covert (eg, unconscious bias). Theresa May commissioned a review in 2016 and the facts show that the UK has move very little since the MacPherson's Report (ref Stephen Lawrence). The Guardian also did a series during the latter part of last year collecting even more data than is found in May's report. The data was so upsetting I could only read it in small bits. Racism is very much alive. It has never gone away. It's now embedded in our institutions, the way we perceive and treat BME people, etc.
(2) It is also worrying when people say they don't see race. It is of course nonsense. Not seeing someone's race is trying to ignore a fundamental part of who they are and the experiences they have had. It's very important to see race. The thing is to see it, understand how being of a different race affects our experiences in terms of how society views and treats us, and ensure we are not part of the problem in perpetuating injustices.
(3) Children do see race. As the only BME in the school, your daughter will be aware of it. She will want to be like her classmates. It's not fun standing out and being the other. More so if you are BME. As someone pointed out, white people have not been an oppressed race or considered "the other" or "the lesser" and that makes a big, big difference in terms of standing out.
OP, I don't face the problem you are grappling with. However, in your shoes I wouldn't put a BME child in this situation. For me, the problem would be the impact on her self-image. At that age kids do not want to stand out. They want to be like their friends. Wear the hair like their friends. Look like their friends, etc.
She may not face overt racism, in terms of name calling but it is very likely she will feel, at the very least, awkward. Find a more diverse school if you can.