Hi
I had a similar problem with NPower and in the end, apart from bombarding them with phone calls & emails, I contacted the regulator. As soon as I mentioned that to them they panicked. It took 9 months but eventually my bombarding them paid off. If an energy company provides you with both gas & electric and has the option of paperless billing the excuse "that BGs system will only allow ONE account to be registered online" to me, is complete and utter rubbish. Even their website indicates you can view both accounts. Someone has been trying to fob you off it would seem. Simple: they can't advertise a service if in reality they can't provide it!!
My concern with NPower was that I didn't know what my bills were and didn't want to be stung at the end of the quarter with a hefty bill.
If I were you I'd put it in writing and send it by special delivery which is guaranteed: you will be able to track when it was delivered. They then have a certain amount of time to respond and rectify. I would also state that if they don't resolve to your satisfaction that you would be getting the regulator involved. I also have it in the back of my mind that if they don't respond within so many days (can't remember how many) you would be entitled to some compensation. (Could have sworn I saw it on Watchdog the other week). Also, energy providers are dropping their prices so you could also threaten them with moving your custom elsewhere.
The ball is in your court now!!
Good luck!