Just had the chance to make a test using FeedWordPress. When processing the feed, the plugin returns the shortcode output and that is what gets imported into the websites using FeedWordPress. This is why you see in editor the div with all that information on videos in it. This could be solved by checking if output is feed and instead of showing the HTML output just use the shortcode again. There are a few problems with this though.
The shortcode uses the post id for the imported video. When processing the video shortcode, the plugin extracts all video information stored on video post in order to output the video. Since the website doesn’t have the same videos imported as the original one, this may lead to inconsistencies. Also, on your other website you would need to use the plugin again to output the video which means you must purchase another copy of it according to Envato Licensing.
Will try to find a solution for this but it might take a while, thanks for understanding.