I need to be brutally honest. I think you are NUTS for wanting her in the room.
What she's going thru is nothing new for many children. I do not think having her in the room is the answer. In fact, i think it could do the exact opposite.
1. Your husband has anxiety attacks - he is not going to be equipped to deal with you, the baby and your daugther. Giving him one more responsibility and person to worry about is not going to help him.
2. You really want to be having a baby AND worrying about if your other daughter is okay thru the entire process? Whether your pushing or on a surgical table, you really want to be looking over to her and asking her, "Are you okay honey?" Focus on your job as a pregnant mother which is: Having the baby.
3. I don't know about you, but I was on the verge of puking after every push. I don't really care to see ANYONE puke or on the verge of puking as an adult. That's just one difficult scenario. What if you are having contractions and in great pain - it kills me to see anyone in pain as an adult, it would absolutely be frightening for a child to see their mother in that situation.
4. Or worse - heaven forbid, something goes wrong with you or the baby. She will have to witness serious stuff, and get shoved aside - which is already her worst nightmare.
5. Which brings me to my last point - I am almost certain most doctors, nurses, and hospitals want as few people in the room as possible for good reason. Even if they allow it, what makes you believe that you, your husband or any of those doctors and nurses will pay attention to her? You are going to be focussed (hopefully) on having the baby and she could very well be shoved in a corner and ignored. Is that really the message you want to un-intentionally give her?
If she's insecure about the baby - this would not be the first time in the history of babies. Enlist a trusted family member or family friend she is close to and see if they will assist in making sure she is comfortable at the hospital, she can confide in, and they can talk to her about all these feelings she's having. Maybe someone outside your immediate family circle can help her see she's a major part of this and she will not be forgotten.