what can be done for a circularprocessindicator to show incorrect input registration?

You can try this

if (_registerFormKey.currentState!.validate()) {
    User? user = await FireAuth.registerUsingEmailPassword(
                       name: _nameTextController.text,
                       email: _emailTextController.text,
                       password:_passwordTextController.text,);

    setState(() {
        isProcessing = false;
    });

    if (user != null) {
        Navigator.of(context).pushAndRemoveUntil(
            MaterialPageRoute(builder: (context) =>
                ProfilePage(user: user)),ModalRoute.withName('/'),);
        }
} /// add code in this line
  /// if use input wrong
  /// update UI to show the button again
else {
    setState(() {
        isProcessing = false;
    });
}