If the woman who died was quite young – I’m assuming she was – then the funeral is likely to be very well attended. There could be over 100 people there. You and your daughter are not going to stand out, as you said you will sit quietly at the back and she knows how to behave. The family may not even notice you were there until the end.
As an adult, I took great comfort from seeing people at my parents’ funerals who were old friends. Some of them didn’t even come to the wake, but I knew they had been. Going forward, it might be helpful for your daughters’ friend to know she also remembers the funeral.
If it’s an open invitation, I would definitely go to the reception afterwards even if only briefly. The friend might be delighted to see someone of her own age. You’ll judge it for yourself.
I don’t see why the funeral should distress her, other than if she sees her friend upset. There will be readings, music, probably a lot of laughter. Much of it will go over your DD’s head. Death is part of life. So what if she sees adults crying? Hiding grief and pretending it’s not happening is not a great preparation for adult life.
I wouldn’t bother the family, they’ve got enough to think about. They will be pleased to see you there showing your respect for your friend.