IMO...in order for the pups to be registered at all, both parents must be registered with the same registry.
If the registration 'application' was never sent in and it is the puppy papers you still have on your female, you cannot register the litter until she is registered herself.
Is the male also APR? If he is a different registry the pups cannot be registered.
Personally with it being APR, I wouldnt waste my time registerong the litter and just sell them without papers on spay/neuter contracts.
Since your girl jumped the fence, you are responsible for all the vet bill sincurred for these puppies...including the puppy you may choose to give to the owner of the male.
You, in good faith, since you have to live next door to him should offer him a stud fee equal to the price of a puppy, or simply allow him to choose a puppy and take it. He can then choose to keep it or sell it himself to recooperate a stud fee.
Once pups are weaned, use the money made from the adoption fees on the puppies and have your female spayed.
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