Thanks again, Gardyloo, for the very helpful ideas. You are a legend!
This weekend is a long weekend in London, so we will be away for a few days. But I will look closely at the itin you have suggested to see if we can fit it in.
I will check out AS website for availability first, then call BAEC (am gold so hopefully they will waive the booking fee?) to see if they can book using avios.