I don't think you're going to find a reputable breeder that's going to have a dog ready for you in four weeks either. Usually they have waiting lists. I got my rescue because I um... didn't want to wait long enough for my breeder's bitch to go into heat and then for the puppies to be born and grow up. 
You say you don't want a terrier but then you list a westie as a possible breed? What do you like about westies that makes them different from the terriers you don't want?
Another lhasa's a good option. You could also look into bichon frises, shih tzus, and havanese.
You say you don't want a terrier but then you list a westie as a possible breed? What do you like about westies that makes them different from the terriers you don't want?
Another lhasa's a good option. You could also look into bichon frises, shih tzus, and havanese.