At the elementary school my DCs attended (again in the US) there were whole class toileting breaks to age 6 and after that if anyone needed to go they simply put up a hand folded into a fist and would be nodded or gestured by the teacher to take one of the three hall passes. If all three were in use then students had to wait until one was returned.
No need for a verbal interruption. No need to reveal personal information. Lesson went on without fuss of any kind.
When I was in school I avoided the loos at lunchtime as there were always certain cliques hanging out in them. Kirsty, is it possible the poor girl in your class has similar reservations about using the loos at lunch?
Did it occur to you that the girl might have known her loo use would take more than the time she had left to use it at the end of lunch? Children can get constipated, especially if they don't drink much during the day out of fear of having to pee and being made to wait.
The girl should have told the teacher she had her period
For real?
Someone with your approach would be drummed out of both the (excellent) schools my DCs attended.