I understand the dilemma and think it would be good to send her a private message on Facebook that only she will see. Give her your email or phone number, and tell her you would like her to connect with you so you can talk. This way you are not asking her for information and she can reply to your request if she wants.
You could also just explain in a private message that you are going to be at a particular coffee shop at a particular time and if she is around you would like to see her. This gives her a chance to respond — and you don’t have to put yourself in the position of asking for a date. You are going to the coffee shop anyway and you are asking her to join you.
But, either way, let her know you are interested in connecting with her.