I currently have a rank 9 healer chocobo, but I want the DPS barding, would it be possible to -let's say- when the bird is almost rank 10 (1-2 kill exp to get to 10) reset my SP, put all points into DPS, get rank 10 and the DPS barding and then switch back to healer chocobo?

Or another way to say it: Do you retain the first barding you obtain if you reset your horsebird's SP?