Both my sons have allergies diagnosed by a professional allergist and do fine with Basenjis. We have just got our 3rd B two weeks ago and no sneezing, coughing, itchy eyes. I think for you that it MUST be something the dog is bringing in or perhaps your dog is one that performs one particular way and my family lucked out with 3 of the other way. Maybe someday they will have a test like the Fanconi that shows the DNA of this family tree has more dander and the DNA of that heritage has less dander to near none.
I do agree with the smart person that has posted on this site the best idea I have ever heard: rub a t-shirt all over the possible family addition, sleep with the t-shirt, and THEN decide. I know it is too late for you but a good thing to repost for readers.
Good luck with your situation.