This feels like another anti-competitive/monopoly-type play from Apple, but this is unclear to me:
Apps that exclusively use a third-party or social login service (such as Facebook Login, Google Sign-In, Sign in with Twitter, Sign In with LinkedIn, Login with Amazon, or WeChat Login) to set up or authenticate the user’s primary account with the app must also offer Sign in with Apple as an equivalent option.
So if I offer FB connect AND offer sign in with username/password, it isn't exclusive then, is it?
Anyway, back to your question -- it's all OAUTH isn't it? So make a button that looks like theirs and then launch the proper URL in the browser?