I was supposed to be alone for my birth. My OH refused to schedule off any bit from work, or even tell them he might have to leave etc. He works weeks at a time out of town. He's within state, but sometimes up to 4 hours away. And this caused most of our fights during my pregnancy. I was so beyond terrified; I have no direct family here, no close friends. I was eventually having weekly panic attacks at the thought of going into labor. I had no one to drive me, no one to be with me, no one to help with afterwards. OH said I could drive myself or if I was going to whine about it, then take an ambulance.
(I do not have another child, so I'm not sure from that angle.)
But 2 days before my due date OH was in town for a couple of hours. And I randomly went into labor in those hours! He was beyond terrible during my labor and kept telling me it wasn't real contractions, and me getting worked up was why it hurt. And he didn't believe me until my water broke in the bed...
I was beyond terrified and anxious to be alone, and I thought him being there would help. But he just demeaned me the whole time. He dragged me down the stairs by my arm; he went 80mph down the interstate. When I arrived 2 hours after labor began, I was already 7cm. And by the time I was 8cm I kicked him out
None of this is exactly like your situation except to say that the nursing staff was there for me. They deal with people by themselves frequently. I was trying so hard to make sure my daughter was okay, and to focus on my body, that it was easier with less people in there.
Personally I would try to find another hospital who will allow your child. That's insane. I have known too many women who want their birth to be a family experience. This includes children of any age. I lot of times it is "policy" to not eat food, or to do this or that, but it doesn't not mean they can force you. If you want your child, and most importantly you partner to be with you, they can't stop you. And maybe then find someone who is more understanding. My midwife knew I was probably going I be alone and was sad about it. I've seen nurses look after children. They are there to help you. Like maybe one can be in the lobby with child? So you can have someone with you? Or maybe you can hire a sitter?