I think I figured out the easiest work around for this. Do a 301 URL redirect as described here: https://support.squarespace.com/hc/en-us/articles/205815308-URL-redirects#toc-format-redirects
In summary, you go to settings -> advanced -> URL Mappings -> add this to the box (obviously replacing "product1" with your products URL:
/shop/product1 -> /shop/p/product1 301
Then your old link will just redirect to the new URL.
This doesn't get rid of it like the original poster was hoping for but can help avoid having to make a bunch of changes to old links (or physical QR codes like the second poster).