I agree that your daughter should not be doing anything excessive that might damage her health or further risking your son's health.
Tick.
I agree that you should feel annoyed that the school is putting pressure on you to allow her to play this game.
Tick.
However .... and here is the but ....
Maybe the school is putting pressure on you to allow her to play because they want her to get her colours and this match is critical in that.
Maybe to qualify she has to play this game and the school knows how devastated she will be if she misses it and does not get her colours.
Maybe the qualifier for colours is quite rigid i.e. play x get y, no discussions or compromise, often these things are like that, or we would all have colours for something.
I think you are fully in your rights to not allow your daughter to play.
However with that comes consequences, and in this case she might not be well enough to play which means she might miss this important game which means she will not be awarded the colours.
I can't imagine they are using this to persecute you - I think they want your daughter to be awarded colours and realise that this match is the defining point to make that happen.
If she is not on the match list she will not qualify, the school is not going to lie and put her on the list if she is not there.
She plays = colours.
She does not play = no colours.
Simple, but really crappy.