Sometimes Dogs just DO NOT like each other. If you find a dog take your B with you and do proper introductions.
I had a male and a female and the male tried to kill the female on a regular basis.

My boy's grandmother was 24 years old - In Sweden

Go to another Vet! This is NOT normal !

Breeders in your area are a good start and keep trying with BRAT. Remember they are volunteers and lead other lives.
Go to Dog Shows and keep trying.

I read your reply and keep trying or contact someone here-
https://www.basenji.org/joomla/index.php?option=com_contact&view=category&catid=75&Itemid=97

Have you tried breeders in your area? They sometimes look for homes for retired dogs.

And what happened with BRAT?

Get some books- they help. Basenjis are not like normal dogs and you need to be careful with punishments

Is he still cleaning himself?
MIne stopped cleaning his groin and it looks like that unless I clean it with ear wax cleaner.

Try Agility. A Basenji isn't a dog who will just sit on the couch all day-they need to run and you need to keep the mind busy.Idle paws are the Devil's workshop!

With my Male-he was a Rescue- I lay on the floor with a paper bag of treats and some potato chips. I had my back to him and ate the chips with loud munching sounds. Basenjis are soooo nosey and he soon cam and stood in front of me. I move away and made sure my back was to him. After a few times he came right up to my face and I gave him a treat. Make sure you use something only for that- a special treat he cannot resist.
Sitting on the floor and doing the same thing works. Try not to touch him right away.
When you give him a treat with one hand gently touch hime with the other. It may take some time but he has to learn that you are a good thing.