Hi,
For android phone, you do need Google’s FCM system and Google Play Services support. I don't think there's a way to legally use Google FCM push in China mainland.
iPhone seems has no this kind of issue.
Authenticator for Android in the public cloud in China
https://support.microsoft.com/en-us/account-billing/authenticator-for-android-in-the-public-cloud-in-china-ebbef05c-a429-4236-8570-1bb1900fec35