Honestly, I do understand your concern, we all worry about our dcs and whether or not we're doing the right thing by them, particularly relating to their health, but if she's no worse she really doesn't need to be seen in A&E. If the OOH GP had been at all worried about her, he would have told you to forget the show and take her to hospital there and then, he obviously thought - as do I, that if she's well enough to perform in a show, no matter how stoic she is in nature, the situation is obviously not an emergency.
Being worried about missing school isn't a good reason for using emergency care services, she only needs to be seen by them if she's getting worse or has new worrying symptoms that have arisen since the last time she saw a doctor.
In some ways, I think your dh has a point, the first of the two diagnoses is a benign illness that will get better on its own. Yes, she will be in pain until she's better, but its not dangerous or life-threatening in any way. He is however, being pretty insensitive. Obviously you didn't know what was wrong a few days ago and were doing your best to work it out - you're not a doctor and could only speculate based on her symptoms while the medics weren't listening to you. The very fact that this doctor has listened, taken you seriously and come up with a potential diagnosis is vindication of your feelings and actions.
Take her back to the GP tomorrow, talk it all through, explain what the other doctor said and ask for him to arrange for her to be seen at the hospital to rule out appendicitis.