I'm fairly certain that detecting someone hiding their e-mail from you and then making them pick a different e-mail goes against the spirit, if not the rules, of Sign In with Apple.
That said, it would be extremely beneficial to pop up a screen saying "Hey, is this the e-mail you want to use for communications?" and let the user decide.
That said, removing third-party sign-in is also a fine solution, almost definitely a better one, and simplifies things immensely for everyone involved (assuming their sign-in form in the app supports saving passwords to the keychain).