No, they should not be allowed to stay.
By their own claim they are the other sex. So they ought to deal with everything that comes with that. They can't have it both ways.
That said, if the alternative is to have actual males in a girls' school because they "feel like girls", I prefer that the trans pupils stay in the school for their actual sex.
I'd prefer schools to be separated into schools for females, schools for non-females, and co-ed.
I used to be in favour of letting girls stay in girls's schools as that'd allow them to change their mind about the gendernonsense, but I have since realized that most probably won't do that until they are in their thirties, if at all, and until then, they are best kept away from girls who signed up for a school where they would not be subjected to male violence, biased teaching and all the other shit..
While transwomen have just the same violence levels as other males, apparently transmen manage to trans their behaviour, too, and are more violent than other females.
An all-girls school won't do much good when it is the transed children who get 70% of the teacher's attention, and only 30% are left to the lowly cisgirls.
And obviously that is the thing that would happen if the teachers actually see the transed child as male, as the transed child and the transed child's parents will demand.