I don't know much about the ILP# (AKC) or PEN# (CKC) process so I can't really comment on whether or not this status could be obtained on Izzy.
Another rescue I volunteer with in Ottawa brought in a Golden Retriever without papers. The woman who adopted him got a CKC performance event number (PEN), and is doing competition obedience. All you need is a vet to agree on the breed.
From the CKC website:
A PEN number allows an unregisterable dog of a CKC recognized breed to participate in competitive events that are appropriate for the breed. To apply for a PEN, you must complete a CKC Performance Event Number Application and submit the form along with the following requirements:
1. Two recent and clear photographs of the dog in standing position, one full frontal view and one full profile. The photographs are to be signed and dated by a veterinarian confirming type of breed and size of dog. (i.e., 20" Golden Retriever.)
2. A veterinary certificate confirming that the dog has been spayed or neutered.
3. A statement as to why the dog cannot be properly registered with the CKC.
4. The dog must be identified with a CKC approved microchip.
5. If the dog is co-owned, all co-owners must sign the application.
6. The appropriate fee, as noted in the CKC Fee Schedule.