Rowing mom, I am sure you know this but as my orthodontist explained to me, braces create an inflammation to allow for the expansion and movement of teeth and jaw into the right place. My son sounds very similar to your DD. He also started ticcing a ton with dental work. Poor kids! As far as antimicrobials I don't think we ever took any but we are a mediterranean household and add garlic to everything so that, I'm sure, helps.
I think that, while he was in his big flare, many things would trigger him, specially sugar (immediately) and gluten. Not sure about milk. As he starting to steady himself he was able to incorporate those back. Olive oil is our base fat. Fish oil, which seems helpful to many kids was terrible for his tics. If his tics were speially bad Motrin always helped. It was clear to me that I needed to tackle the inflammation. Good luck Rowing Mom!