Contact PRS Customer Service. They can arrange to sell you backplates that are not normally available on the website shop - I've had to this when working on lefty Cores.
You can even try to get another one signed and sent to you...I purchased a Chuck Levin's Anniversary issue 594 guitar at Levin's PRS clinic, but could not remove the backplate in time for Paul to sign it. The store and PRS street team arranged to have a signed 594 backplate sent to my home.
If you know the exact place/date where he signed it, that would surely help ("Paul signed my McCarty 594 backplate at the Chuck Levin's clinic on November 21, 2019..."). They might just hook you up. Also, I would buy two if that happens...one plate for the guitar and the signed plate saved to protection (or mounted in a shadow box, perhaps).
Good luck!