Well what a silly doctor. I am not 100% sure. (i had one when i was 16) It is just a mouth swap, you, the baby, and the father. I think my father instigated it. maybe call another doctors surgury. I am pretty sure that both you and the father will have to split the costs 50/50. But if either of you are on a low income you can get it covered by medicare, so worth looking into. call a helpline and ask. or maybe call the local hospital. goodluck sorry im not much help!
EDIT: SCAT you reminded me of something! For my paternity test my Father wanted it done (so it was elective on his part) that is why my mum got it free (she is low income) I remember her saying if SHE was the one to instigate it she would have to pay her half herself. so yeah... SCATs right!