As for Singer, I doubt he'd even bring the Hellfire Club into the series since a group of aristocratic Mutants with a secret society manipulating industries and governments would be seen as "too far out" for his X-Men movies and how the evil mutants are characterized solely by Magneto's terrorists. Dark Phoenix would be boring as hell if it was just Jean going nuts for the sake of plot.
I think you're assuming too much.
Dude, the audience and critics alike would complain that secret societies who dress up in outfits and manipulate events from the shadows belong in James Bond and not in Singer's X-Men. I'm starting to think most fans of the X-Men films are snobby types who are ashamed of the fact that X-Men began as a comic and are contemptuous of the term "Superhero".
Same with "Dark Knight" fans, I mean even Nolan himself sometimes seems like a guy whose ashamed that Batman is a comic character and says stuff like "Dark Knight isn't a superhero film, it's a down-to-earth crime drama!". There's no room for the fantastic with these folks.