I can't tell you a particular brand because, usually, there is a specific ingredient that the child is allergic to. My son has had terrible reactions to sunscreen since he was about 9. We knew there was one sunscreen that had not caused problems so we tried to figure out which ingredient was not present in it but in the ones that had caused the rash and swelling. I think it is Avobenzone that he is allergic to. The first time he had a reaction, we had put a "for faces" sunscreen on him. His eyes swelled shut, he had a heavy red rash all over the area it had been applied and his face felt like shoe leather. It took days of taking benadril for it to go away. We tested a few different ones on his arms before discovering his culprit.
It can also be fragrences used in the product that can cause a reaction for some, but if you are using sunscreen for sensitive skin, it may not have that. Sensitive skin just may not be the problem. It may be an actual allergic reaction to a particular ingredient. I would look closely at the ingredients in what they were using and try to find one with different ingredients. Test it on a small area first. Also, sometimes the ingredient does not cause the reaction until the sun hits it, so make sure you try on a small area that will be exposed to the sun. Also, there may only be one ingredient different but if that is the one that is the problem, it only takes one.
It can be a real pain but if it is an allergy, it is worth it to find out what. Allergies can become worse over time, which is what I think happened with my son. He now refuses sunscreen from other people and we use the same brand every time to make it simpler. Some people are allergic to sun exposure but you might look at the sunscreen angle first. Good luck.