For some reason, fully-flex business class seats seem to be cheap from Prague at the last minute.
Still not *that* cheap but it seems to be a good place to start.
In general, the two key techniques are:
1. Use ITA matrix far and wide - I would suggest
LHR, DUB, CPH, SVG, GOT, PRG, KRK, AMS, BRU, LUX, DUS, BCN, MIL, NCE, HEL
as your starting point and adjust accordingly for schedules.
Add /ALLIANCE ONEWORLD and/or have LHR as the change point to focus on BA and AA-operated flights.
2. Add a hotel or car to see if that opens cheaper bucket availability, although this tends to work better in WTP or WT.
3. WTP on a day flight isn't so bad.
Searching specifically on Saturday 27th to Friday 4th, the best I can then do is GOT - LHR - IAH - LHR - DUB for £2,169.