As an employee I find it frustrating myself when I'm trying to issue a form/letter or do something in a customers file and the system just wont allow it...I can only imagine how frustrating it must be for you, especially when there's money involved!!
What I think would be the best thing to do is to have your friend write a letter - Centrelink need to be satisfied that you are separated by verifying with a 3rd party - the letter will need to say when you separated from your partner, her relationship to you and/or your partner, whether she's aware of any shared financial commitments/assets, arrangements with care of the kids, who has changed address...as much information as she can that confirms you are no longer with your partner - ask her if she could put her phone number on the letter, and tell the Centrelink staff that they are welcome to call your friend if they need further information - hopefully the person you get will think laterally. I think the letter that is sent to referees to verify a separation is actually a proforma that asks specific questions - perhaps if your friend's letter doesn't quite cover anything, the staff member can view the letter on your file, call your friend and ask them over the phone. I'm not sure whether this will be possible but maybe it will - be insistent, and if it doesn't work, call the Customer Relations Unit who can be really good at getting an outcome for you when you've done all you can and its just not working! All the best!