He mentioned that he was signed off and allowed to drive the vehicle.. therefore insurance is still valid. Yes they know he was speeding but if he wasn't charge then it has no bearing with insurance companies. Whichever party is charged are the ones liable for all damages in the case of insurance payouts.
In NSW if you do not receive a sticker you have 14 days to drive around wherever the hell you feel like - if not cleared after this time you may only drive the vehicle to a point of repair & home as well as to get it cleared. Surely its the same in most states?
I would bet that neither party were charged which is why the other insurance company is trying to get him to admit liability. If they had been charged there is no way they would of issued the letter of demand.
I spent 6 months getting the run around from my own insurance company years ago - I was hit at a 110km on the F3 whilst stationary and pushed into the car in front. The car in front thought that they had sustained no damage (tow ball) and took off before I could get their details. Person who hit me was uninsured so I went through my insurer - they tried to tell me that the $1100 damage to the front of my car was the result of an accident after the one that had happened (hit up the ass) and prior to be loaded onto a tow truck 20minutes later...wtf? It took months of ringing every day to finally get my money - I always wonder what damage did happen to that other car as hitting the towball tends to twist a few things underneath!
Either give it too your insurance company, bend over & cop it, or hire a traffic lawyer - they are the only options.