I think it's probably the high fat content, the additives (e.g. nitrates) or a combination. At McDonald's, there's nothing else to balance out the meal like vegetables or fruit - so it's fatty burgers, fatty nuggets, and fatty fries. Same with the hotdogs. Also the white bread in the burger and hot dog rolls has a lot of junk in it. Your daughter may have a digestive imbalance that makes it hard for her to digest it - I don't know if she eats that at home with no problem, but I raise the issue just in case.
Regarding the post about it being soy, it's not soy itself but the additives or whatever it is washed in - pure soy is nature's protein and has been eaten for 5000 years by cultures all over the world. A lot of soy products in the US are tainted by the processing methods. Before you go crazy eliminating all kinds of foods from her diet, or getting a huge amount of testing in one so little, try controlling the quality of the items she eats. I know that can be hard when you don't know the sources.
You could also consider Now for Kids nutritional supplement which will establish a healthy balance in her digestive system, give her great nutrition, and make it far less likely that she will react to those foods in the future. I can help you with more info and connections to parents whose kids benefited and no longer had "food allergies". There are very few true food allergies - these reactions are on the rise because we get far less nutrition in this country than we did 20-30-40 years ago. When we were kids, everyone ate PB sandwiches on white bread, and no one had peanut allergies, milk allergies, gluten allergies, and so on. This huge epidemic of allergies is due to changes in our nutrition, including processed foods, fewer whole grains, and fruits/vegies grown in depleted soils and picked early to be shipped for weeks on end. We are missing key elements (phytonutrients, trace elements, vitamins, minerals, etc.) and just not able to digest things the right way. Processed vitamins don't help much either as they are not absorbed more than about 25-35%. So you might want to look at a better supplement to help eliminate this problem in your daughter.