A good vet is a good vet even if they aren't "basenji experts"! The good ones will be willing to learn about the breed but at the end of the day, a basenji is still a dog. ANY dog can be quirky!! My vets never saw basenjis before mine and I've had zero issues. They are patient, kind people who care about their patients. (My dogs are show dogs and very easy patients though! used to standing still while someone examines them). A good vet will be willing to listen to you!
Ask local friends where they go and get a few recommendations. REMEMBER!!!!!!! It's 100% OK to decide a vet isn't the right fit. There is no shame in shopping around! Don't keep going to a place you don't love. It's up to you to train your new baby to be OK with being handled/examined. Work on playing with paws, opening their mouth, gentle restraint, etc. I also belive that muzzle training is a good idea. If your dog (any breed!!!!) is ever hurt/painful they might try to bite the vet OR you so having them used to a muzzle is a wonderful thing. Make it positive so going to the vet is no big deal!
Good Luck with your baby!