If you can't tell your friend the truth, blame it on the doctor.
Say in case of an Emergency they only want the husband or birthing coach in the room. If she is deadset on coming tell her how much it would mean to you to have her stay with your first child so you wouldn't have to worry....
I did this with my mom. I love her, but she gets crazy scared in a medical situation. She was with our oldest son when I was having our baby 2 years ago.
I know a part of her wanted to be in the room and see everything, since she missed the actual birth of our first son (c-section only one allowed in the room).....but I could live without one extra person seeing my girl parts this time around!
Best Regards,
Chrystal