Air France + Delta via CDG and LAX £2817 for travel in August
BA + Hawaian Club + First via LAX £3164
Have also found BA/AA First on BA and AA for £3478 this seems like a reasonable fare
_ LON BA X/LAX BA HNL 2392.03AAP9UK BA X/LAX BA LON 2392.03AAP9UK NUC 4784.06 END ROE 0.634398 XT 120.00GB 30.63UB 3.40YC 4.30XY 3.10XA 20.00US 4.50AY 255.00YQ 2.80XF LAX4.50
However, and as usual, if you begin in Europe, Amsterdam for example then Club on BA/AA starts at around £2000.
Fare construction INCEU28
AMS BA X/LON BA X/LAX AA HNL 1270.61INCEU28 AA X/LAX BA X/LON BA AMS 1403.25RRCEU28 NUC 2673.86 END ROE 0.753963 XT 39.32UB 10.90CJ 12.80RN 1.70VV 3.40YC 4.30XY 3.10XA 20.00US 4.50AY 223.80YQ 2.80XF LAX4.50