I don't know which part of the EU/non EU Europe you're going to as you don't say.
I think your parents in law are being really difficult.

American processed food has GM ingredients and banned additives so you won't find it in the EU unless reformulated. Nor will you find American cheese because of the bovine growth hormones. However you will find, all over Europe a huge variety of food. It won't all be additive free unless organic and in Italy there will be hectares of pasta.
In the UK you'll find dairy free milk, cheese and yoghurt and gluten free stuff no probs. Almost all supermarkets now have a Free From section. Look for Lactofree, alpro and Provamel brands too. In London there are organic supermarkets eg Whole Foods, Planet Organic, Daylesford Organic and you'll find organic food at Waitrose, Sainsbury's, Tesco and in more limited amounts at ASDA, Morrisons and Lidl.
GM ingredients have to be labelled so do irradiated foods and as Europeans don't like either you probably won't see any.
You'll find ghee in tins in the bigger branches of Waitrose, Sainsbury's, Tesco, ASDA, Morrisons and in branches with big ethnic populations as well as in Indian/Muslim stores.
I don't think it's worth making your own snacks in case they are confiscated at customs.