I gotta admit, I'm a bit ashamed right now.
Thank you, Mr. Gimbel. The problem was indeed an incorrect password. While we did indeed enter the new password into the PSE, we did not delete the entries first. Instead we simply tried to update the keytab entries with sappsegen -p SAPSNCSKERB.pse -nopsegen -a svc-sap-sso@GENERIC.DOMAIN.
Should anyone find this post: Delete the keytab entries before entering an entry with an updated password.