Help my son is allergic to our new puppy.

Hello everyone,
i have just joined this wonderful forum in hope to get a bit if advice on allergies to dogs.I brought a puppy 3 weeks ago,a pedigree black labrador who is 12 weeks old now.Everything was going fine until last week my 5 year old son developed a dry excema type rash on his back and shoulder but no other symptoms.I took him to my gp who ordered some skin prick tests and this confirmed that my son has a mild allergic reaction to dogs.We desperately dont want to get rid of our puppy and i have been advised to buy hepa air purifers and a product called biolife petal clense and home clense that proclaims to reduce the allergens on our puppy by 90%.Also i have brought 2 doggy gates so the puppy cannot play in my sons bedroom or playroom,plus i have stripped all the bedding and cleaned all rooms throughly whilst hoovering twice a day to.Hopefully this will all help but im wondering if anyone else has a child that is allergic to dogs as any advice or further information on treatments ect would be fantastic.
Thankyou so much x