No BTDT unfortunately because I am not vegetarian, but ultimately it's definitely possible to do an ED and be vegetarian. Plenty of vegan women (so no animal products AT ALL) have healthy pregnancies and BF with no ill effects. It would just be important for her to get calcium/vit D and protein from other sources. So, depending on how *strict* she is, she could have fish (especially salmon/sardines, bone in for calcium), eggs, lentils, coconut milk, fortified rice or soy milk, etc.
I do agree though that to know for sure she would have to do a full elimination. Simply "cutting back" really wouldn't tell her for sure one way or the other - some babies are so sensitive that they react to even trace amounts of dairy, while others can tolerate when mom eats cheese or yogurt, just not straight milk. The fastest way to know for sure is to fully eliminate all dairy (including trace/hidden) for a minimum of 2 weeks (better if one month, takes 2 weeks to completely eliminate protein from breastmilk, another 2 weeks to completely eliminate from lo's system). Some moms notice improvement right away, others it takes longer. If she notices improvement, she could try reintroducing trace/hidden and see if it's tolerated. (And if it is she could go further and try to reintroduce yogurt or cheese if she wants to, etc.)
Especially given that she is vegetarian, if she's going to do a strict elimination I would consult with a dietitian - just to make sure that she's getting all the nutrition that she needs