What about another cookie name?
Snicker Doodle=Snickers or Doodle
Sugar
Spritz
Oatmeal Raisin=Rae or something with Rae in it-Sweet Baby Rae?
Chocolate Chip=Coco
Monster Cookie=Monster
Sugar 'n' Spice=Clove(r)
Or, what about just Cookie?
You could also go with the "Ginger" as something "spicy" and call the new pup something in that theme:
Red Belle Pepper=Belle or Pepper
Lil' Hot Sauce=Cayenne
Dash o' Hot Sauce=Tabasco or Tabby, Tabitha, Dash
Zing or Zingy
Bamboo Shoot=Bambi?
Ok..I'm stretching it now!