Kids & pet allergies

Recently our dog of 10 years who has for most of that time been an outside only pet has been spending time inside. Due to our current housing situation there is really no good way around this. She has a rug & that is her spot and she is only allowed on her rug while inside. My 1 year old loves to play with her and is constantly rubbing all of her & her rug. Unfortunately, he is breaking out in hives 😭! We bought some special cleaning spray and dog wash that is supposed to help eliminate allergens, & though it seems better, he’s still breaking out. I’m thinking it might be her saliva that’s an issue & not dander... 🤷🏻‍♀️. Has anyone dealt with this and found a good solution that wasn’t giving up their pet?

Hopefully I don’t get any hate here, but just to clarify ... we’ve been in this situation about 1 week and are actively trying to find a solution. I am not putting my pet ahead of my child’s health. Also I have a large breed dog who enjoys being outside, we spend lots of time playing with her outdoors & she has always had appropriate shelter for all weather conditions.