K.I.
What about just adding your last name to their last name? Like a hyphen situation? That way they will have your name and you can probably phase out using the dad's last name...I mean they would still 'legally' have his last name but they would also 'legally' have your last name also so that way you would be able to say their last name is the same as yours as well? That might work? I do not think you need the father's consent to add to the name?!
You should just go down to the SS admin office and ask...it's way cheaper and you shouldn't need any thing from the father! Make sure you ask which last name is actually 'THE' last name...like if the dad's name is Smith and your is Jones ask which way it works, should they be 'Smith-Jones' or 'Jones-Smith'? I have heard sometimes it's the first one and sometimes it's the last one...so just go in and ask...
That way I bet you could register them for school and Dr/'s appt. and such and just give your last name and it would work...