Another point.
We have been told over and over than trans children are very shy about their bodies and prefer to change separately.
For that to happen they would need to give a reason surely?
In the case they are in communal changing then of course the other children will notice that they are a different sex.
The other point is that we are told often that trans children have particular vulnerabilities and needs.
If a child moved schools once socially transitioned I'd think any parent would be alerting the school about their child's trans status so they could keep an eye out for issues. Whether related to MH, bullying etc.
And you can't change your sex legally until you're 18 here I don't know about USA. Any school would want docs for a child they will be enrolling surely.
How things work in real life do often not seem to be given a thought.
In practice you'd need a situation where
A child socially transitioned and then changed schools probably to a different area where no one knew them
With no docs and the school would accept that
The parents would lie about the child's sex to the school/ not disclose their trans status
The child 'passes' as their acquired sex and no one notices any secondary sexual characteristics/ that they are the opposite sex when changing
The school do not ask for records from previous school
Probably more
Also in practice the two trans girls who are good at sports in USA are openly trans and there have been lots of articles etc about how supportive the response has been.