I was a little anemic at the end of my first two pregnancies. I was giving birth in a hospital anyway, and there were no interventions, not even pitocin for the afterbirth. After my first daughter was born I was bleeding pretty heavily for maybe 15 minutes (hard to say, just a rough estimate), my midwife was concerned and started talking about maybe cauterizing, I freaked out and refused. She said, "well, go urinate and see if that stops it. Sometimes if your bladder is full it prevents your uterus from contracting properly." I went and peed and sure enough I stopped bleeding heavily within a few minutes. After my second daughter was born nothing went wrong at all.
Happy, healthy births can definitely happen at a hospital, and low hemoglobin shouldn't make a difference. They almost certainly just want you to have quick access to blood transfusions in the unlikely event that you do have a problem. That's a good thing, so even though I know you are giving up your ideal birth and you will need to grieve, I think you should try to look objectively at it: many women used to die after birth because of things like bleeding, nowadays we can stop that, your caregivers just want to protect your health.
You almost certainly will be just fine and not need any kind of intervention, but you can rest assured that if you do you will be in the right place to get it.