The easiest way is to befriend a GGL with a spare partner card slot. The second easiest is to fly. Long flights (over 2000 miles) attract 140 TPs a pop in premium cabins, but booking a three-sector short run for 120 TPs can be quicker and cheaper. F@210 from the west coast is basically limited to SFO/LAX/SNA-JFK and LAX-MIA.

Codeshares can’t be booked in the US without a longhaul connection. It doesn’t have to be the UK - any AA/IB flight to Europe would do it (AA flight needs to be on BA code but IB is ok as IB) - but either way you’ll have to leave the US. You may find BA’s short intra-Caribbean flights easier to access, though potentially not cheaper.