Nancy, I totally disagree with your comment. When people place a dog and pay the dog's bills, they aren't going to have problems getting it back. That isn't in anyone's book "abandonment." Sure there are jerks out there, but not usually. Also, having worked in rescue for nearly 40 yrs, much as our egos don't want to believe it, dogs adjust to new homes. And they adjust to coming home. Show dogs may go out on the road for a very long time, and they are fine coming home.
Just going to be very blunt. Choosing a dog based on being "hypoallergenic" ,often if not usually , results in a failed home. They don't exist, period. I was glad to see you understand the issue is nonshedding. Yes, nonshedding dogs leave less dander. But until you actually have a dog in your house, you aren't going to know. Your best bet is to find a rescue that will allow you to foster with intent to adopt. That way, if it doesn't work out, not a huge crisis.
All of that said, if you haven't had experience with a basenji, please talk to local rescue and spend some time reading the boards. These are not the dog for casual owners who haven't researched the breed. I have found them much more challenging than Rottweilers and Chows!