I fully advocate comprehensive third party. If I didn't have full third party, I'd park my car up.
Then probably have it hit by somebody anyway, which seems the case with you. (I'll give you the benefit of the doubt and pretend you weren't driving around without it.)
To start with, I'd call them up (AAMI) again for a few reasons. Firstly, make sure they are fully aware you're in contact with the owner of the other vehicle and that he has informed you everything is in order. They should be able to tell you more than "Not approved". Especially since he can.
On that note, I'd also be asking them what the time frame was to have your car fixed. Quotes as Boiracer suggested is also a good idea, simply just to have that knowledge for future reference.
Let us know how you go tomorrow anyway. There is no reason AAMI shouldn't be helping you out A.S.A.P.