I have to say that just reading the headline my initial thought was: "no, a Basenji is not for you". But Shar Pei's are, IMO, not the easiest dogs to deal with. Plus you have something of a pet menagerie, the maintenance of which requires no small amount of pet skills. So my guess is that you would be fine.
The great thing is that if you get an older dog from a reliable breeder you will have a very good idea of what you're getting. Puppies are more hit and miss in that it's hard to tell at 12 weeks how they'll turn out. So if you can find a Basenji which has the right personality you might have a good situation, enjoyable for both you and the dog.
My only question would be the cats and birds. Put crudely, Basenjis are killers. That would be something to consider.