'butter but not milk (aren't they the same).'
Butter is fat, without milk protein or lactose. Milk has both milk protein and lactose.
Going off certain foods temporarily to see if symptoms improve is one way to determine if an allergy or intolerance exists. It's not the same thing as cutting out food groups indefinitely or some sort of fad diet. In the case of nasal symptoms, dairy is often the culprit. As long as adequate calcium is available from another source, cutting out dairy, for instance, does not necessarily lead to problems. My DCs have been dairy free all their lives. The runny noses (and accompanying post-nasal drip, coughing, stuffed noses at bedtime and wakefulness during the night) are not a problem.
There are lots and lots of posts on the Allergies Topic wrt food allergies and therapies, methods of investigation, remedies, etc. A common theme is the frustration of a lot of parents whose doctors seem to know very little about food allergies. The science is still a developing one, it seems, with lots of grey areas, and what works for one child might not work for another.
As far as the TV goes, having been in the position of the mum who was told by one DD's friend that her DD wasn't allowed to watch TV while she was playing with mine, in my house, where my other DCs were happily watching whatever was on, I have to sympathise with you on the TV issue. Here I would say, your house, your rules. But in the case of the allergy thing, ask the ex for some reading material, spend a bit of time on the Allergy Topic here, and also ask the ex for a bit of advance warning (a few days at least) on what's on and what's off so you can plan food for the two days out of fourteen, as NonnoMum says, that the child is with you.