Send data from one WP instance to another and process it there

I solved it with mmm’s hint and created a simple custom REST endpoint.

WP1 (Frontend, sending the data):

function send_form_data_to_backend($form_values) {

    // add secret-key
    $form_values['secret_key'] = "abcde123";

    $url="http://backend.example.com/wp-json/example/v1/save/";
    $response = wp_safe_remote_post( $url, array(
        'method' => 'POST',
        'timeout' => 15,
        'redirection' => 5,
        'blocking' => true,
        'headers' => array(),
        'body' => $form_values,
        'cookies' => array()
        )
    );

    if ( is_wp_error( $response ) ) {
       $error_message = $response->get_error_message();
       echo "Something went wrong: $error_message";
    } else {
       echo 'Response:<pre>';
       print_r( $response );
       echo '</pre>';
    }
}

WP2 (Backend, receiving the data):

add_action( 'rest_api_init', 'post_request' );

function post_request() {

    register_rest_route( 'example/v1', '/save/', array(
                'methods' => 'POST',
                'callback' => 'rest_function')
    );
}


/*
 * REST Function
 */
function rest_function( $data ) {

    if ( $data['secret_key'] == "abcde123" ) {

        $msg = example_main_form_processing( $data );

        return $msg;

    } else {
        return "Wrong Key.";
    }

}