Dear Speccie,
This is hard for you as well as your friend and the baby's dad.
I think a large part of the problem is that the medical professionals, be they supporting a home birth or a hospital one, do not really know the mum, and sadly some of them are not trained fully.
I have been in almost the same situation as you. A close friend planned a home birth. I'd had an easy hospital birth and was born at home myself and was full of the idea that all women can do it. She was very keen. Two weeks overdue the domino midwife recommened she go to hospital for a scan. My relaxed friend waited a day, then went in. I think her baby and her were FAR more scared than a stranger can notice.
She had a scan and there was merconium. They recommended induction and she agreed. I was with her until this point and then I left (I had to go home to breastfeed by baby - I wish I'd said 'oh lets start her on formula). I also wish I had said to the staff 'she's not quite as strong as she seems.'
They induced my friend, moving swiftly to a drip. After a few hours the midwife said she was 7cm. The midwife, sadly, was a trainee, and was wrong. My friend was not dilated at all.
My friend was in her mid-thirties and it was her first child.
She should not have been given only a trainee midwife.
A consultant finally came. He said she needed an emergency C-section. Over an hour later they did the operation. The baby, a girl, was deprived of oxygen and was born brain dead. She died four days later.
My friend was very brave.
She also sued the hosiptal and managed to change their policy, regarding the way the midwives tell what is the baby's skull when they are testing dilation, (and what is not).
My friend and I continued being close for quite a while. It was hard for me too, with a 9 month old baby girl. We don't see each other anymore. Perhaps she doesn't want to be the girl whose baby died. She has not had another.
Four and a half years later I had a son. I think I was too scared before. I went to a hospital abroad to do it. They were great.
I learned a few things from all this, and if I were to be involved at all in someone elses birthplan I would say:
If you are two weeks overdue don't even bother with induction. Just have a C-section.
Always get more than one person to feel for dilation on your first birth.
It's best not to be too rigid about birth.
A card is good. Just listen to your friend. Share your own theories with other people, not her. Take care of yourself too.