I think if you validate her behaviour, you're making a rod for your own back, it's likely to cause further shit storms between the teacher and child.
She was angry, not unconsolable, doesn't that tell you the punishment had its intended effect, your DD will be reluctant to repeat said behaviour, plus others may also think twice about being the class clown.
If you validate her behaviour, you're saying that she was in effect wrongly punished.
You stated they've had run ins in the past, possibly a clash of egos?
If I offer a different perspective, I was high achieving at school, it irked me, even though I was from a council estate, that such ill mannered children were disrupting my education. You've had the teachers POV, I feel it necessary to say that it could have been a pupils favourite lesson, they wanted to get as much from it as they could, due to your daughters behaviour valuable time was lost.
You stated yourself that she can be 'lippy' and this teacher has said, enough, I won't tolerate it, continue to act out I will continue to highlight your lack of co-operation on the matter.
Another example, DS doesn't get on with a certain teacher, DSis didn't like her that much either, so DH goes all guns blazing validating his behaviour. He's a bit younger than your DD so they expect a bit of immaturity, but he has this teacher an hour a week, there's always a clash because DH/DSis/Nephew think said teacher is draconian. I feel he has to learn whether you're madly in love with your teacher or hate them with a passion, they have a job to do. Yes being the class clown is funny to surrounding mates, but I think from the perspective of pupils who lose out due to his behaviour, plus the poor teacher.
I do believe teachers have a difficult time in the first place, over worked and underpaid. I know a teacher and for a seemingly strong person, the stress has affected her physical health. I also believe parents add to the stress when they go, 'Oh my little snowflake should be treated this way.' Then you have the children exhibit a 'you can't touch me' persona, which adds to an existing burden.
My final point is, your daughter is sporty, she was held back from participating, she was given what the teacher saw as an appropriate punishment. Yes the lads found it entertaining, but I doubt a boy would be treated different, their punishment during class is likely to be more humiliating. She is angry, some would suggest that's the desired effect to nip bad behaviour in the bud. If your daughter wasn't sporty and hated PE, I could understand why as a parent you'd be upset, but it appears her ego is merely bruised.
Speak to the form tutor, ask about her behaviour in general during other lessons, if this is isolated, mark it as a clash of personalities and move on. Should your daughter be unfairly targeted, then you have reason to be hopping mad. Write it off this once, should there be a repeat where your daughter isn't to blame, by all means take a different approach.
Just don't let your DD think she can behaves how she wishes. Regardless of whether she's doing the school a favour being in teams. Advocate that in life there are times you have to agree to disagree. This is one of those times.
I hope I don't come across too harsh, we look at teachers if unaware or ignorant, thinking great working hours, many holidays. The truth is it's a profession where being salaried, you can put in 100 hours a week it makes no difference to pay. If you want to be a good teacher, you have to go the extra mile.
I wasn't sporty for obvious reasons, I hated PE, there was one teacher who initially I thought was a twat, but when we found something I was good at she was full of praise. No parental intervention, just me gritting my teeth and getting on with it.
Good luck at lunchtime.