That is indeed a problem. Apple Support will require a Proof of Purchase for the original version, before they provide a replacement serial number.
See: Pro Application Replacement Serial Numbers
Have you tried, if Apple Support will accept the evidence of a running installation without the receipt to provide a replacement serial number?
Try to argue, that it is not fair to make you buy Aperture again, since it will no longer be developed.
The only way I know of to migrate Aperture to different Mac without serial numbers, would be to update Aperture to a version, that is only sold at the App Store. This would associate Aperture with your AppleID, so you can download it on all Macs you own.
This worked well on Mavericks with the update to Aperture 3.5.1, but the current release is Aperture 3.6, and that requires Yosemite. You would have to upgrade both macs to Yosemite.