If your daughter is very sensitive, lactase pills won't let her eat whatever milk she wants - they'll just delay symptoms from being immediate to showing up later (likely why your daughter had a delay from when she ate the dairy to when her tummy was upset).
There's a difference between a milk allergy and lactose intolerance, so it's good you will be getting her in to be tested. The former is usually due to proteins in milk products, the latter to milk sugar that can't be digested. Milk sugar can't be "cooked" out of a food, nor can it be avoided by switching to goat's milk, etc.
Try cutting out all obvious sources of milk: milk, cheese (especially cream cheese, creamy havarti and dried cheeses like parmesan and romano), dried milk, milk solids, cream, buttermilk, whey, dried whey, and lactose.
Milk products hide in lots of things. Some obvious like salad dressings, muffins, and soups and sauces, but also non-obvious places e.g., in bread and hamburger buns (extends shelf life), pizza crust, cookies, cakes, waffles, canned soups, teriyaki sauces, cup-o-noodles, margarine, batter coatings (like McD's chicken nuggets), dried pasta that has ferrous lactate, and lots of pre-packaged foods. One of the sneakiest things you'll see on an ingredient list is "natural flavoring" or "natural sweetener" which are often euphemisms for milk sugar.
A bigger challenge awaits when your daughter hits school and you have to send separate snacks so your daughter doesn't have to risk milk exposure there. Note they won't let her have lactase pills at school without paperwork filed through the school office.
Your doctor can advise about how to see your daughter gets enough calcium and Vitamin D if she's not drinking milk.