First of all, make sure you and your doctor have the terminology down. Allergies are strictly defined as IgE-mediated reactions to proteins (and some glycoproteins, that is sugar protein complexes). Sensitivity to milk sugar is lactose intolerance. The foods you will need to avoid will depend on which of these your daughter is sensitive to. Some foods with milk sugar added will NOT contain milk protein (i.e. casein) and could be safe to eat if she is allergic, while some foods will have casein but not lactose and can be eaten if she's lactose intolerant.
In general, here's an easy trick: look for foods marked pareve or parve. These are designations for people who keep kosher and indicate that there are no dairy products (protein or sugars) at all in a product. These can be hard to find in a regular supermarket, but can be found in abundance at a kosher supermarket. Sadly, it looks like the only kosher supermarket in Honolulu just closed in October, but they still appear to be running some kind of mail order service (see oahukosher.com). Most health- and natural food stores are also likely to carry a greater amount of nondairy products as they tend to be more allergen-conscious.
Otherwise, stick to fresh fruits and veges and unprocessed foods as much as possible. You may have to resort to doing a lot of your own baking. Margarine can be substituted for butter. Soy milk, rice milk, and Mocha Mix (the only REAL nondairy creamer with no dairy) can be substituted for milk. Mocha Mix gives the best results in things like puddings because it is higher in fat.
As for goat's milk -- be careful with this. Many children who are allergic to cow's milk will cross react with goat's milk. If your daughter is not subject to strong reactions (no hives, edema, vomiting or danger of anaphylaxis) then you can probably safely try out goat's milk on your own. Otherwise, I wouldn't risk it, unless at the doctor's office or within proximity of an ER and an epipen.
As for oral challenge as the gold standard for food allergy (known as DBPCFC or double blind placebo controlled food challenge), that is changing. Food challenges are expensive and can be dangerous, and several studies by key researchers in the allergy field have shown that blood tests for specific IgE have equivalent diagnostic accuracy to food challenges and skin prick testing. See especially those published by Hugh Sampson in the US, (Sampson HA. J Allergy Clin Immunol. 2001; 107(5):891-6), Sampson HA. Allergy. 2005;60(suppl. 79):19-24), Christina Cobbaert in the Netherlands(Cobbaert CM, et al. Clin Chem Lab Med. 2005;43(7):772-81), Marcus Ollert in Germany Ollert M, et al. Clin Chem. 2005 Jul;51(7):1241-9), and Sara Prates in Portugal (Prates S, et al. In vitro methods for specific IgE detection in cow's milk allergy. Allergol Immunopathol (Madr). 2006 Jan-Feb;34(1):27-31). You can download free abstracts of these from pubmed.com, but will otherwise have to pay for the full articles. I helped to write/edit that last three articles, so I might be able to dig up pdf reprints someplace (I might have Hugh's papers, too). I can email these to you if you like, and also a couple of easier to understand review articles I've coauthored.
Adoption of blood tests by US docs is finally gaining greater acceptance (Europe is way ahead of us on this, as they are for adoption of most medical advances), so sadly many kids are still subjected to the 100 yr-old practice of skin prick testing (a great revenue source for allergists, not fun for the kiddies or their parents).
BTW, the good news is that most children diagnosed with a food allergy before the age of 5 are likely to grow out of the allergy. The bad news is that food allergies can mark the beginning of the allergy march, which can progress to inhalant allergies (think molds and seasonal rhinitis) with an endpoint of asthma. However, strict avoidance of diagnosed allergens can short circuit the allergy march and prevent these endpoints.
Oh yes, and stay away from NAET. I'm sure someone has recommended it. It's pure bull-hooey and is based on no scientific methodology or physiological reality. It "diagnoses" "allergy" at a rate that is up to 60 times higher than it's actual prevalence, and then, no surprise "cures" what isn't there in the first place. I've seen some of the studies posted in clinicaltrials.gov. They do not meet CONSORT standards for quality evidence-based medicine methods of research and none has ever been published in an actual peer-reviewed journal. All have been self published, only, with no oversight or opportunity for critique by recognized experts in the field. The reputable journal publication process tries to ensure that studies are well founded and properly interpreted. I'd say that anyone who bypasses this process has something fishy goin on.
Go to aaaai.org for a good source of allergy information.