Add "Site Search" to context menu for selected text for websites listed as search engines

IMDB is added to my Site Search entries, so I can search the site by typing imdb in the address bar and pressing Tab. However, when I’m on that site and I select some text, the right-click context menu offers to search that text in Brave but not on IMDB. When I’m on the page for Edgar Wright and I select Simon Pegg’s name, it would be very useful to be able to search for Simon Pegg on IMDB!

In general, when a user is on a site which the browser knows how to search, there ought to be an option for it here…