Another possibility might be Rat Terrier. They do come in brindle, although that colour is out of favour. There is one in my neighbourhood that has a bit of a Basenji look to him, and his tail is very short.......not sure if it's natural or docked.
If you want to know anything about basenjis before you adopt one, please feel free to ask anything at all. We have a tremendous amount of collective experience and knowledge here.