I would not bother with allergy testing - HV is right that allergies change, but also because intolerance is just as common (prob MORE common) with dairy, and intolerances can not be identified in allergy tests. (Allergies are reactions by the immune system, you can draw blood, test in other ways, etc to see if there is a reaction. Intolerances are an inability of the digestive system (bowel) to break down the food (usually proteins) and the only way to test for certain is a bowel biopsy, which just will not happen, not that you would want to!)
The way that food issues are most commonly "diagnosed" in kids is clinical - which means that you remove the suspected food, and watch for symptoms to change. If symptoms disappear then that is enough confirmation that there is an issue with that food. Sometimes there are reasons to do a "challenge" which would be to then add the item back into the diet and see if the symptoms return as a further "proof" that the questioned food was in fact the problem, but usually it's not done right away...the thought is that if it is an allergy the reaction can be more severe and if it is an intolerance you are just adding more pain/discomfort/damage to the bowel.
At Henry's age I would not cut dairy for just 2 weeks and expect drastic improvements - it can take at least 14 days for any remnants of the milk proteins in his body to disappear completely and then time after that for the body to heal from the exposure to milk and symptoms to disappear. It could happen much sooner, but if he still has symptoms after 2 weeks I would not take that to mean it's not dairy iyswim. I would personally aim 4 weeks minimum at that age before you could know for sure - the longer he has been having the milk the more damage potentially done to the intestines. And, remember it would not just be milk, cheese, yogurt, but also cut out all hidden milk as well, or you just will not get an accurate picture - so no skim milk powder which is commonly in breads or biscuits, whey, etc etc. There are lists posted of all ingredients to avoid (I used to have one and am sure I have posted it here in the past but not sure where) and it is quite a long list when you start reading it - milk crops up in different forms in many places!
When your HV says that he will get colds and things and make it harder to notice, I can see where she is coming from but if he does have an intolerance then it's more than just a runny nose that you will notice...ie/ you say he has circles under his eyes, but he wouldn't have those with a cold. And when you have a cold, you know he will feel run down, coughing, and so on, and it's more likely that he caught it from somewhere, rather than just a constant runny nose, right? So saying that it's cold season might mean you don't notice the difference is a bit silly to me....a cold lasts 7-10 days not much longer!
HTH?