Well it also depends on the support people at BT, there are some people there that know stuff, but there are a lot of them that don't.
I am not say they are wrong, but if the modem is syncing fine but just not authenticating then the issue in my mind is between the exchange and their network.
I have also had an issue in the past with a similar issue and when I reported it I was told, oh its the hub, we will send you a new one out, it arrives a couple of days later to result in the same issue.
So phoned BT again and was told oh its the hub, we will send you a new one out, where I told them no, you already sent me a new one it wasn't that, then they say its internal wiring, I said no its not due to its syncing up fine with the equipment at the exchange.
And several hours later the fault was at the exchange that connects to BT's network.
So BT support staff often read from a queue card, this is why they ask you all the questions.
This was why I asked if you have tried other modems/ routers etc to rule out hardware your end.
I might even be wrong, but I am sure more people will comment with more things it could be.
But yeah it was bad timing for it to all go wrong.
Paul