Apologies your post isn’t terribly clear.
Am I right in thinking that due to medical treatment you won’t be able to pick your child up from school?
Because you don’t have any other child care options she is going to spend the day with your MIL instead, missing school entirely?
Tbh, although I sympathise with your position I can see why they didn’t authorise without more information.
Why can’t your MIL spend the day at your house, dropping off and picking up your DD?
Is there no school friend who might kindly invite her to play after school as a favour to help you out?
Why is your DD’s father unable to assist?
You don’t have to give every single detail of your life but you need to give them enough info to understand the situation.
If you’ve sent a rude email then yes, YABU I’m afraid. They aren’t interested in your life. They are interested in your DD’s wellbeyand education.