Baby daddy

Hi! I have a kinda weird question. Please dont judge..I am 17, Me and my babys bio father are no longer together, and he is not involved at all. However, I am with another man now, who plans to be there with me during delivery. I know they will ask me to name the father, but I dont want to since he isnt involved at all. My question is, Will the hospital pressure me on it if I say hes not involved? do I have to name him? And will they judge me for having a man who isnt the father with me? Thank you so much!!