I'm going to ask that our friend change into nicer clothes before joining us for dinner
What is wrong with her clothes?
I'm afraid that my mom will judge her for wearing a short dress.
I get it, but it's not nice to tell your friends how to dress. You should let them dress how they want.
I can try to talk to my friend and tell her that mother is a bit conservative and might get offended by her clothes, but I will not force her to dress differently.
I think your idea is a good one. It's okay to let her know that your mother is more conservative about dressing for dinner, but then let your friend decide what she will wear.