I would go with AA, and I also would suggest looking at some minor changes in routing that would get you more miles and let you avoid Heathrow. If you ticketed it through AA in Stockholm rather than BA, you'd also probably avoid some additional fuel surcharges and fees. The AA general sales agent in Sweden (TAL Aviation) is very competent.

ARN-HEL-JFK-YUL-LAX-DFW-LIM-EZE-SCL-IPC-PPT-AKL-SYD-DRW-PER-SIN-NRT-HKG-DEL-HEL-ARN, 20 coupons and 19 "counting" flights. (The PPT-AKL sector is not eligible in the OW ticket but the "surface" segment has to be counted. However, Air Tahiti Nui, which operates the PPT-AKL flight, is an AA partner and would earn full miles in L class.)

Finnair also earns 1 AA mile per flown mile in L class, so worth the switch from BA in my opinion. HEL is a much easier airport to navigate than LHR as well. By switching your route to DEL instead of BOM in India (then use low cost local carriers to move around the subcontinent) you can avoid another open-jaw, and just return to Helsinki on Finnair's nonstop, again avoiding Heathrow.

Regarding mileage earning, if you take the AA codeshares on ARN-HEL-JFK, use the US segments including the transcon, and then fly to Lima on AA from Dallas rather than on Lan from LAX, and if it's less than 3 months from leaving Sweden before you reach Lima, then you could enroll for AA's Platinum Challenge, and receive 100% bonus miles on some carriers once you've reached 10,000 qualifying points, which I estimate would occur on your DFW-LIM flight. You'd also get lounge access and other benefits as Platinums.

Counting the eligible bonuses that would follow, I make your total mileage yield on AA to be in the neighborhood of 50,000 miles for the whole trip, i.e. enough for a transatlantic award or one to India or the middle east, or a couple within Europe. Might be worth the effort to amend your itinerary.