I don’t think they’ll remove the “Fox” name from legacy credits of movies they now own (they haven’t here). It’s only for future projects that they’re changing the name. For example, the poster for The French Dispatch already says Searchlight Pictures on it.

This is similar to how Ted Turner bought out MGM’s entire back catalogue in the mid-80’s and, even after his assets were in turn acquired by Time Warner in the 90’s, the legacy credits on the back of the Warner Bros dvd releases of those films still said Metro-Goldwyn-Mayer just as they always had (2001: A Space Odyssey is a good example). Also worth noting is that while the opening logos of those films remained unaltered, the turner logo was added to the packaging since the copyright technically belonged to that company now and not MGM.