The transits through London (if under 24h) wouldn't attract UK air passenger duty, so the big fuel fine numbers are mainly BA for the segments flown (TLV-LHR, LHR-LAS and LHR-TLV.) You might also be hit for YQ on SIN-LHR regardless if it's BA or QF metal.

If it's cheaper online, then you could go ahead and book it there. The days of AA's RTW team ignoring BA fuel surcharges are long gone.

Otherwise I don't think your taxes/fees total is terribly unusual.