Can a dog registered with the Continental Kennel Club be registered with the American Kennel Club?

Not sure of the question but I think your asking if you can breed an APRI and and AKC dog together and get papers on the puppies?
The answer is YES. You can go a lot of different directions with registration.
You can dual register both of your dogs with ACR - American Canine Registry - http://www.americancanineregistry.com and then apply for litter papers on the puppies.