I think you'll still have difficulty with centrelink & CSA. If the child isn't your partners, you'll need to claim CSA officially from the bio father, or you will only be entitled to minimum FTB payments. I really don't think the name will have much to do with anything.
C'link will assume the baby is your partners, I guess, but if he doesn't want his name on the birth cert, will he be ok with that?

Like I said, you'll need to speak someone who knows a bit more about that side of things.
I agree.
I am going through something similar ATM in relation to DS. He had my last name which I recently changed (and we're different from DD)

Best to speak to someone at both CSA and centrelink to get advice in regards to their protocols