How you handle this depends a lot on what your birthday is like. If it's something really intimate at home with three friends, obviously you don't have to invite coworkers. But if it's something bigger, like a dinner or something at a public place where several colleagues are going, that's different because she'll definitely find out and it could seem weird.
The practical trick that actually works is inviting her casually and without drama, but making the nature of the event clear. Something like "hey, I'm having dinner on Saturday, do you want to come?" without making a big deal out of it or acting like it's something special you're offering her. That way you're not at fault for not inviting her, she has the option to come or not without pressure, and if she doesn't show up or declines, there's no tension because it was a laid-back invitation. The important thing is that she knows the possibility existed and that you weren't specifically excluding her.
If your birthday is something more private and closed off, just don't invite her and that's it. You don't need to give explanations or feel guilty about it. It's totally normal to get along with someone but have them not be part of your inner circle, and she understands that perfectly even if she doesn't say it out loud. Most people have coworkers like that and don't stress over it.