You don’t build up an immunity. Lowering the histamine load with drugs can help a bit, but the sure fire solution here is avoidance.
I don’t know about not building up a tolerance (not immunity). I’m horribly allergic to cats, but after almost years with the monster herd (a mamma who gave us 4 kittens that we kept) I’m doing better.
1) lots of vacuuming
2) limit where the dog goes, especially to the child’s bedroom. Have areas of he house where the child can go that are dog hair free.
Very limited exposure, with amounts building over time. They are now treating kids with peanut allergies by giving them peanuts every day, starting with extremely small amounts that gradually increase over time. (according to my allergy doctor.)
I think that is untrue. I think it would be true to say that some allergies cannot be mitigated. But not all and certainly not in children.
Many children outgrow childhood allergies.
I have had some experiences with that personally, but everyone does not react the same or have the same reactions.
Recently there was a paper on the peanut allergy that as most know can be fatal. In this case they introduced peanuts to a child in very small micro amounts under medical supervision. They actually were able to fix the problem.
Works in pets too. I have a weinerdog that was allergic to a particular flea chemical. But without it, in the south, he was being eaten by them as well as mites..So finally I had to do something so I went to the farm supply store and bought a bottle of the chemical. I diluted it into a spray using 1/8 teaspoon to 10 ounces of water...sprayed him, and no reaction....raised the amount to 1/4 teas. and I got a hive breakout so I backed it off to the 1/8 and continued for two weeks...spraying him everyday. Long story short, within 3 months I had increased the dose to the level of a flea bath mix and then applied the monthly flea application that is 50% Permethrine, and he now has no problem with the chemical..
Having said all that, I would limit the contact with the puppy. Just having the dog in the house should give enough exposure and if that is too much, you might think about not having a pet until he stops showing allergy signs. As the child will certainly come into contact with many pets over time...and some Diphenhydromine would be helpful. (benadril) But be careful with it....it will put then to sleep....they have a dose for kids, or did....(long time since having kids around)
I built up immunity with cats. My dog is hypoallergenic so no problem with him. But, cats used to make me unable to breathe and it was awful. After a time it subsided.