So the topic of balances being due later doesn’t appear to be true for my booking. Like the OP, I have a small (£400) deposit for a £3k balance that was due on 9 April and is still due on 9 April. The departing flight is on 14 May.
There are reminders in my calendar to call BA Holidays to discuss with them at the last possible minute. I don’t want to waste the £400 deposit, but I also don’t want to throw good money after bad.
The situation won’t be resolved by the departure date, neither here nor in Greece. I would really love to be on holiday but have come to appreciate that this simply isn’t going to be possible...