Have not had a problem or know anyone with that type reaction but heard it does happen.
My suggestions: Brush daily, which you are doing. Vacuum way more often, daily if you have to. Dander or flaky skin happens during shed. You might try a good brushing then a lukewarm bath with sometime like Earth Bath (for dogs) using a Zoom Groom in the bath, then outside in the sun (a windy day helps) and more brushing. The bath helps loosen the undercoat which is shedding. Keep up the brushing and vacuuming. Definitely consult your pediatrician about your child's reaction.