I would do the rest of the dog as regular needle turn & then make a bias strip for the tail & sew that on separately. I remember this flower I got from Desiree's Designs that I absolutely loved, but spent I don't know how many hours trying to get that silly "lollipop" swirl in the center by needle turning 1/16". A guild member finally suggested I make a bias strip for it & applique that down and that worked perfectly!
-- Just tuck the raw end of the bias strip under the backside of the dog & stitch the tail down first, then the dog's behind.
