There's a difference between lactose intolerance and a milk protein allergy. Intolerance is different than an allergy.
With lactose intolerance, the small intestine is not making enough (sometimes not any at all) lactase, which is the enzyme that digests lactase aka the sugars found in dairy/milk/cream.
With an allergy, the body is actually allergic to the milk protein aka casein/whey in the dairy.
For a successful test for lactose intolerance and dairy allergy you need to do a true elimination diet. You have to read labels and make sure there's not a single trace of any lactose, lactate, lactylate, casein, caseinate, whey, milkfat, butterfat, cream, powdered milk, cheese powder, condensed milk, anything that related to milk. Nothing. Remove dairy completely from her diet for a full 4 weeks. That means absolutely no dairy, not even the lactose-free versions. It takes the body two full weeks to eliminate dairy products and to be free of them. Then the body needs two additional weeks to see what it feels like dairy free.
After those four weeks, if you reintroduce dairy with a glass of milk and she has diarrhea or constipation with stomach upset, serious gas, pain in her bowels, then it's likely lactose intolerance.
If she vomits it back up, gets eczema or a rash or shows other signs of allergy, then it's an allergic reaction. But there could be signs of damage in her stomach that would need to be seen with one of those little scopes.
It could be combination of both. If she gets a reaction either way, stop giving her dairy. Period. Call a pediatric gastroenterologist at your local Children's Hospital and request a breath test to determine lactose intolerance for sure. Also request a milk allergy test which they would do at a different time. They will also set you up with a pediatric nutritionist.
My middle daughter is lactose intolerant. We've BTDT.