You need 61,205 EQP, which at 1.5 EQP/Mi in J or F means around 41K BIS Mi.

Here's one possible itinerary that gets you there -

LAX AA xLHR AA MCT BA LHR AY HEL AY CDG CX HKG CX BKK CX HKG CX SIN AA xNRT AA DFW AA SJU AA DFW AA BOS AA MIA AA LAX.

That comes to almost 44K miles, good for almost 66K EQP, so you could edit it a little and still be safe.

For mileage earning, you'd want to use the AA codeshare on LHR-MCT, AY to and from HEL, and the AA code on the JL flight between SIN and NRT, in order to keep your 100% plat bonus, which isn't granted on BA or JL flight numbers. Doing so along with cabin bonus should net you right around 100,000 spendable miles for the trip.

Remember no more than 2 stopovers in the continent of origin, and you can't leave the US once you return (hence the SJU run instead of some longer segs to Central America.) Also don't forget the advance purchase requirement and the minimum travel time (10 days from first to last international segment.)