"The only media I've seen him watch/watched with him has been: The Walk (Movie 2015), Interstellar, Once Upon a Time (TV), I Love Lucy, Modern Family, Scream Queens. Not sure that really helps lol."
Well, A Gentleman's Guide to Love and Murder is like Scream Queens in a way.
And Fun Home and Spring Awakening sort of have the family drama of Once Upon a Time, Interstellar, and Modern Family. In a way.
Honestly, knock out AAIP and just pick whatever one you want. It's your birthday. You should do something you enjoy. Whatever you pick, he's going to see a great Broadway show and if he doesn't love it you can always make it up to him the next time you make plans. :)