Regardless of where you buy it, the LONE4 option doesn't really work for you because of the second of those two rules - (1) no transit through point of origin, and (2) no second entry into continent of origin unless one is a transit without stopover. Even originating in some other country than the UK wouldn't help because of the second rule, even if the first was moot.

However if you buy a Global Explorer ticket somewhere besides the UK, the second entry rule doesn't apply, only the point of origin transit rule does.

As with business-class ONExs, Sweden is relatively cheap for LGLOBx originations. It would make your route ARN-LHR-EBB-LHR-CAI-AMM-DXB-HKG-NRT-LAX-LHR-ARN, which comes in just over 30K miles, so you'd need to look at the LGLOB34 ticket. The price of the LGLOB34 bought in Sweden today is around US$320 more than if bought in the UK (and around $1200 cheaper than if bought in the USA.) Plus, you could work with the AA office in Sweden and possibly avoid some fuel surcharges and nuisance fees levied by BA for tickets sold in the UK.

If you could substitute Vancouver for Los Angeles, ARN-LHR-EBB-LHR-CAI-AMM-DXB-HKG-NRT-YVR-LHR-ARN comes in at under 29K miles, so an LGLOB29 would work, around US$600 cheaper than the 34K mile ticket if bought in Sweden. You could certainly fly separately between Vancouver and LAX and back for less than that.

The Global Explorer would also let you use Gulf Air, so you could go via Bahrain instead of Amman if you chose.

Remember to factor in your access cost (to get to and from ARN for example) into your cost calculations - sometimes the net savings don't warrant the hassle of starting somewhere else.

Edited to add: Disregard; can't backtrack to Europe from Africa on this one either.