Firstly, if you were to look at cash fares you will be hit by not having a Saturday night stay. You may find adding a car or hotel may getting you a much better rate. ITA Matrix is showing a 1 stop in J at £3,830 with AA from LHR whereas the direct BA flight is over £6k per person. An ex-EU will offer better value with various options at less that £3k per person! Going earlier, i.e. on the Saturday would see fares from around £2,100 per person.

If you want to use your points and 2-4-1 then you may have a lot less choice given that this is bank holiday weekend and school half-term.

I have heard, but have no experience of this, but availability for reward bookings can improve a few days before the flights. Might be possible to book something now and then change nearer to the date.
Good luck.