I am in the market for a low allergy/non shedding dog.What would you recommend?

First off allergies to dogs and cats are breed specific.-- The ones that tend to cause the least amount of reactions are the hair breeds-- They are as follows: Shih-Tzu, Lhasa-Apso, Silkie,Yorkie, Bichon-Frise (arguably the best), Maltese , and poodle. Some people with allergies find that terriers cause them no problems (such as a schnauzer--miniature to giant sized) the terriers do technically shed. Most of the dogs above can be over 10 pounds-- Just depends on breeding. The best way to find out what will work for you is to go to breeders, play with the type you are looking at, then go home and see if you react-- There is no guarantee you won't develop problems, but at least if you react after a few hours-- you'll know to stay away from that breed.
P.S.-- the dogs listed in your post are mixed breeds-- It's a new trend. Make sure you don't get a dog that sheds mixed with a poodle such as a labra-doodle-- You'll likely have shedding and allergy problems