Well, a week later, and my efforts to engage with BTC support staff have gone from fruitless to infuriating.  They just keep telling me that my current BTC balance isn't enough to be paid out.  I keep saying I know that, it's the previous balance which claims it was already paid out to a specific address at a specific time, and wasn't, that I'm emailing about.  They just go back to my current balance not being enough.

If anyone needs a reason to stay away, there you go.