My dd was a bit older, just three days shy of 2 when her little brother was born. We had a homebirth, and planned to have her around as long as she was comfortable. We read books and watched youtube videos, and talked a lot about it, but she really didn't "get" it. I wound up going into labour after she was asleep, and the babe was born at 1am. She DID wake up though. My mom was in rocking her through the actual birth.
I wasn't with it enough to notice or care if dd was there or not (insanely fast and intense labour), so people kept her in her own room. I was a really quiet birther with dd, but shocked myself with how loud I was this time around. Also, we had to deal with a shoulder dystocia so there was a bit of a panic as I yelled about ds being stuck. Like I said, dd missed actually seeing the birth, but I feel really bad about the whole thing because she was seriously affected. I don't know if it would have been better if she could have actually seen things or not. She wouldn't look at me or the baby post birth. She wouldn't let me touch her, and she just hid her face in dh's chest. She was really out of sorts and I think she was quite fearful.
If I could go back, I would have her come out while I was in earlier labour to see me and know that I was ok. And then I think I would have gotten someone to take her out for a walk or something once I got hollery.
You absolutely need someone there to look after her while you're labouring/birthing. Is there no one you know who you would trust to entertain her at the birth center and/or take her out of the room if needed? I'm sure many people would be honoured to be asked if your mom isn't there for the actual day!