Stick with AA. They should earn miles on the BA flight in the AAdvantage programme. (although I don't know it well enough to know whether they exclude any particular fare classes). Presumably there are also more non-flying mile earning options in AA for those based in the US...

Since they are unlikely to reach status, or get so many miles as to redeem for travel in Business Class, you'd want to avoid BA Executive Club, where economy redemptions on BA are usually horrible value given the fuel surcharges, taxes, etc. added.