I think you need to re-challenge the ped on that and talk to the supervisor again. I would assume you have all rights, unless your baby is in immediate life threatening danger.
I'd suggest you stay calm, but be firm about the fact you want your daughter discharged that day. I would call the procurator in advance and tell him/her that you want him/her to assist with the discharge of your daughter at he time of your meeting, and to get the ball rolling now. If you get any pushback then yes, get that lawyer right away, maybe make a preliminary phone call to that lawyer to he/she is familiar with your situation and can give the procurator pressure if they are not cooperative.
You know, everyone that I have told this odyssey to have all told me the same thing "but people back in the day (gma, ggma, etc) gave birth at home and nobody died. What's the big deal? They just wanted to take your money".