Would you recommend adding the Video Sitemap to Google Search Console?

You don’t have to do that specifically for the video XML sitemap. It is linked from the Index Sitemap that is generated by the WordPress SEO plugin, which will make Google recognize it automatically. I assume you will have already added that Index sitemap to Google Search Console (former Google Webmaster Tools), if you haven’t, that might be smart to do. You may use this guide to submit the sitemap to Google and other search engines.

Clicking on that will then show you the stats for the individual sitemaps that are contained in it.