Please see https://www.shamimsplugins.com/docs/front-end-pm/customization/send-welcome-message-user-register/

As you want to send message to users when post is published, you can use following code

function fep_cus_user_post_send_messaage($post){

        $author = get_userdata($post->post_author);
        $author_id = $author->ID;
	$post_title = $post->post_title;
	$post_link = get_permalink($post);
	
	
        if ($post->post_type !== 'post')
                return;
        if ( ! function_exists( 'fep_send_message' ) )
                return;
        // Prepare message data
        $message = array(
             'message_title' => 'title', //change with message title
             'message_content' => 'xxx', //change with message content
             'message_to_id' => $author_id
        );
	
        $override = array(
             'post_author' => 1, //change with message sender id
        );

        // Send message
       fep_send_message( $message, $override );      
}
add_action( 'pending_to_publish', 'fep_cus_user_post_send_messaage', 10, 1 );

Please go to Dashboard > Settings > General and check your timezone setup.

I tried in my test installation with shanghai timezone setup and it shows correctly.
Please again check your timezone setup, if require change to UTC and save, again shanghai and save.
Then send a message, and check front end as well as back-end to check time of that message.
Go to Dashboard > Front End PM PRO > All Messages to check that message time in back-end.
If both places show wrong time, Then deactivate all plugins and change theme to default theme and send message and check time.

Let me know.

Please change $time = strtotime( $post->post_date_gmt ); to $time = get_post_time( 'G', true, $post, false ); and see if it working correctly.
Let me know.

Is it working correctly in your setup after this change?