It doesn't sound like you can avoid hurting the friend's feelings. But try to remember that your feelings and the feelings of your mom and sister are what matters in this situation. If you don't hurt the friend's feelings, then you'll be hurting the feelings of yourself and two people you love.
You're probably a very polite young woman, and it's hard to cross over to being an assertive woman for most of us. It's a good skill to learn, though, how to be both polite and assertive at the same time.
I think you should write her or email her and explain that the birthday gift you and your sister are giving your mom is time alone in an exotic place with her two babies. Even tell her you're worried about hurting her (the friend's) feelings, but this is likely to be a once-in-a-lifetime for your mom and you girls, to have time just the three of you like you've planned.
Then insist that NO ONE tell her where you're staying, and that includes the hotel manager. Hotels in Vegas try to be very careful to respect their guests, so that shouldn't be a problem.
Good luck, and hold your head up. You need to do the right thing for your family, not for someone who may be important but is herself being rude and trying to ruin your mom's 60th birthday celebration by making it about HER instead of about your mom and you girls.
Sidebar: my daughter just went through something similar when planning for my 40th birthday, and she told the person, "Well, my mom would love to celebrate with you, so why not let her have two celebrations? You can plan one for her too that way. But this one is just for Mom, Dad, me, and my brother's family."