One thing to note is that if you manage to get access to just the hashes, they won't have sha512(p), so they wouldn't be able to figure out sha512(q). So while

One thing to note is that if you manage to get access to just the hashes, they won't have sha512(p), so they wouldn't be able to figure out sha512(q). So while your are correct that there is (technically) a weakness there, it's not one that's actually exploitable.