He's really cute but definitely not a Basenji. I'm assuming people are using the term mutt to encompass anything not pure bred but many pure breed are crossed with other pure breed to create new 'breeds' , Labradoodle, cockapoo etc etc. This is how new breeds are formed. Yours looks like a lovely little chap, live him for who he is not what he is.
We took in a 12 year old rescue Basenji, a wonderful little chap and very good natured who fitted in easily with our little pack (another male Basenji and a female Lurcher). A few years later whilst on holiday in Wales, Fergus (our rescue) had a funny turn whilst on the beach. We took him to a local vet who diagnosed vestibular disease. We cut our holiday short and took him home and he got through it but forever more his head tilted to the left which gave him a peculiar gait and overall strange appearance, however, it didn't shorten his life and he carried on in his odd looking way to the age of 17.
I don't want to be judgemental, however, on the face of it she took in a 4 year old dog, kept it for ten years, loved it to bits then handed it in to a rescue centre when it became ill and difficult and now wants a puppy. This more equates to owning a second hand car than a dog so I'm afraid I will never understand her reasoning. A dog which you've lived with and loved deserves so much more.
It still seems very strange for you to send your dog to a rescue centre. Did you not consider that that your dog will miss you even more than you miss her. If she was very I'll it may have been best for all to consider euthanasia ! I appreciate that you miss your dog but my original comment still stands - when you let a dog into your life you take on the commitment to look after it for ever. Please don't get another pet unless you can make that commitment.