Profile Page Redirecting to Home Page when logged out

Please try this:

function redirect_to_register_on_profile_page() {
    if (is_page('profile') && !is_user_logged_in()) {
        wp_redirect(home_url('/register'));
        exit;
    }
}
add_action('template_redirect', 'redirect_to_register_on_profile_page');

You can use the code snippet plugin or add the snippet to your theme’s functions.php file

Let us know,

Thanks,