AA & LA are options.
Random sample:
AA963 DFW-SAO 15/Jan in J
50K miles + C$2.70 using AAdvantage
40K miles + C$2.67 using BAEC
LA533 YYZ-JFK 13/Jan in Y, LA 532 JFK-YYZ 20/Jan in Y
40K kms + US$47.14 (no YQ) using LANPass
25K miles + C$48.42 using BAEC
EDIT: I couldn't get AA for ORD-NRT using BAEC, even though AAdvantage showed availability at the lower miles. I'm guessing it's a ba.com glitch that could be worked around by calling in.