It can take a long time to recover from bronchialitis even after the worst symptoms are past. In the beginning they have to prioritise breathing over feeding and really don't feed well. Even after she is able to breathe more easily, loss of appetite is common for a while. Usually babies do better with smaller amounts more frequently.
He is almost at an age to take solids - but I wouldn't take this as a sign that he wanted them. He seems like he is quite allergic. I wouldn't rush to introduce solids. Do you know what he reacts to? Clues with your daughter or other family members may give you a hint