In any case where there's a registration ID system, there's a risk of fraud. If the breeder offers up the ID number to anyone who comes along and asks, it's child's play for a disreputable breeder to pose as an "interested buyer" who simply wants to check the IDs. Once they have the ID numbers in hand, they can then use that information to create counterfeit registration paperwork for their own, unregistered dogs, allowing them to charge a premium price for a "registered purebred" that might be a mutt or have some other factor disqualifying them from registration. An unsuspecting buyer might not even try the ID number, and one who had suspicions and did try to validate it would find that it did, indeed, check out as a real number.
While personally I have no familiarity with the AKC, I have seen exactly this situation happen with other high-priced consumer goods; people foolishly put their ID (serial) numbers on the internet, and disreputable individuals used those to create counterfeit paperwork to pair up with counterfeit goods and attempt to pass them as legitimate. For this exact reason, it is not only perfectly reasonable that the breeder would not just hand over the ID, it's exactly what I would expect them to do.