Ok, an update! Took a whole hour and a half to sort. Strangely I didn't get cut off and thankfully the seats were still available (JFK-LHR)

We had some Avios on FTVs from a part cancelled flight in the summer, but BAH can't access those, so I had to buy 5k to have sufficient. Fortunately they credited straight away, though.

Thanks all for your help.