Make sure the skirt is wool and goes all the way to your shoes. Wear a hat, glasses, and punctuate your speech with plenty of “oh dear”'s, and “my my”'s.
I’d put in a different opinion. A dear friend is suffering from a nosocomial infection. First thing I notice about a Dr. is if what s/he is wearing can be easily laundered. (That, and the size of his/her hands in case they have to be stuffed in any orifice.) I can’t imagine each day’s clothes would be taken to a dry cleaner right away.
Also, don’t wear a fancy manicure or false nails.
Perhaps a shirtdress?
In the UK, GPs and even hospital doctors don’t wear lab coats anymore.
OP, I think your suggestion is fine as long as it isn’t too short. Nice and plain.