You've got plenty of options when it comes to Grell's coat, and it's technically a woman's coat so a lot of fabrics that wouldn't sit well on a man usually come up for grabs. I'd steer away from satins (There's no evidence to suggest the coat is shiny, and matt looks classier anyway) and pick something with a decent weight to it, but otherwise it's your choice. If you can find them in red, suiting or gabardine would be a very appropriate choice. Twill might work if you washed it thoroughly beforehand (Off the roll, twill is usually too heavily starched) but you'd still get quite a stiff finish. If you wanted to go super luxurious, you could choose a good quality velvet. I think when I saw the costumes made for the musical, Grell's coat had a self-herringbone pattern, so textured fabrics are not ruled out either.
Re: historical fabrics, back in the 19th century it was 100% natural fibre, but honestly? Synthetic mixes make our lives easier, are cheaper, don't wrinkle to high heaven, and Kuroshitsuji is painfully inaccurate anyway. Definitely choose a high natural fibre content to get the right look and finish, but period-accurate fabrics means period-accurate upkeep, and I like to wash my outer clothes occasionally.