Dear Strake
Congratulations to you and the future Mrs Strake.
I had a similar situation some years ago (apart from the Mrs Strake bit) with American (supposedly close enough to BA), no C class awards, yadda yadda yadda. Took a while to find out the name of the UK sales mgr, phoned them up in their office, eventually found them in, described the situation pleasantly and professionally. No problem, they sorted the res. It actually goes into the system as an "overbooking", not of the seating in the aircraft but of the fare class (which might have availability zero on such a flight), so you just have to find who has the authority at BA to overbook such a niggardly award allowance that yield management have determined.
Hope this helps as a way to consider going about things.
Good luck to you and Mrs S (future)
PS If you can't spell Qantas properly, no wonder this has happened. Remember your grammar, U after Q except before Antas.