In general, dogs with a high prey drive (specifically terriers, hunting-types, and spitz breeds) will be worse with birds/small animals.
Dogs bred to not take prey (specifically herding breeds) can do better. I'd imagine everything else is going to depend on a the dogs personality, but it might be worth a quick check on some of the other breeds to see what they were bred to do.
Every dog is an individual, and there will be exceptions within every breed. Sight hounds, like your greyhound, that might be fine 95% of the time or herding breeds that didn't get the memo they aren't supposed to eat what they guard.
So I would personally pass on the basenji mix because they are hunting dogs, and judging on their size, probably bred to hunt small prey like a terrier.