If you get a puppy, the cat and a basenji should be fine. An adult B meeting a cat for the first time will typically give chase. Cats run from dogs, typically, and basenji's chase anything that runs away, including people. 
Welcome to the forum.
Sorry you're having difficulty finding a pup. There is an imbalance between people looking for a Basenji puppy and available puppies. Basenjis don't have large litters, there aren't a ton of breeders, and there is only one heat a year. As a consequence the supply is somewhat limited. Plus, as I'm sure you're aware, breeding takes a lot of time, effort, and money. It's a labor of love more than anything.
That said, I'd be quite confident that you'll be able to find a puppy. As I'm sure you know from experience, most conscientious breeders care where their pups land. In your case that's a good thing because there is a chance you'll have your dog compete, which most would find desirable.
As for not being sure what a win looks like, judging is subjective. A loser on day one can win breed and class on day two. (Not only with dogs. I was reading today about how SnapChat has "pretty filters" that always turn eyes blue). It's easier of where the competition is performance based.
