Matt, welcome, and good to see you doing your homework! I am from SE Mich, and can help you out in your search. I only know of one reputable Michigan breeder off the top of my head who has anything right now potentially, but I think even her litter has a waiting list. But there are some potential breeders within driving distance that are also great, so you might still be able to find a good puppy. Most of the time at this point in the breeding season, puppies are usually all spoken for. So you may need to wait until next year to get a good healthy pup from a reliable person.
Where are you located? You're more than welcome to come get to know some Basenjis and I can help fill you in on things you might want to know about the breed. Also, I wouldn't count out a rescue. Sometimes rescue does get puppies, but also some of their adult dogs have already been "tested" around cats and other dogs due to their former homes or foster homes. At least with an adult usually what you see is what you get. So they can help you find a dog that will meet your needs. With a puppy, it is a good way to go if they need to live in those arrangements, but you still can't guarantee that as they mature they will be accepting of the situation. lol
We also have a 15 yr old Schnauzer here that is going through what your Lab is. It's heartbreaking to see them decline, but you know at that point that they've lived a long wonderful life.