You need to put a collar on each barkbark, then get three sets of two bones set outside the barkbark kennel.
When you have done the first set they will come out of the kennel, wait for the lady to drop off an egg and as soon as she has dropped off the egg they will go back into normal mode.
Click on one of each and get them to eat a bone each, and they will romance again, wait for the egg again and then repeat this with the last set of bones.
Wait for the three puppies to go into their cocoon form and sell the three new barkbarks.
The best way so you dont sell your barkbarks that are breeding is to name the two that you use as the breeders.