Regarding alaunt, it is apparently another form of alan, meaning, according to the Shorter Oxford, a large hunting dog, a wolfhound or, in heraldry, a short-eared mastiff. Other variants, already allowed in Chihuahua, are aland and alant. So alaunt will be added to the list.
(Please, no ribald comments about inserting a large hunting dog into a chihuahua.)
I'm not 100% sure about the connection with the people known as the Alans or Alani. The passage Binkie quotes is from a Wikipedia entry that cites no sources, and I haven't seen any other, independent, website with the same account of the origin of the hounds. So it should be treated with caution, even though it seems plausible.