Google flights shows me that $408, but always with a two-stop (LGW, JFK) return and the last leg on AA. Rather pay $433 and have it one-stop both ways all on BA; assuming crediting to AA, you'll get better EQD/EQM earning that way since it'll be distance-based. Also, ITA won't show me the $408 no matter what I tell it...its lowest is always the $433.
Looking at that $433 option though (which is seriously $34 in fare and the rest taxes/fees), there are actually 2 transfers allowed...possible you might be able to build something that goes quite a bit further:
Category 8: Stopover restrictions
NO STOPOVERS PERMITTED.
Category 9: Transfer restrictions
2 TRANSFERS PERMITTED IN EACH DIRECTION FREE
2 FREE IN EUROPE IN EACH DIRECTION.
AND - 2 TRANSFERS PERMITTED IN EACH DIRECTION FREE
1 FREE BETWEEN AA/AY/BA/IB/US AND AS IN LAX/PDX/
SEA IN EACH DIRECTION
1 FREE BETWEEN AA/BA/IB/US AND AM IN MEXICO IN
EACH DIRECTION
1 FREE BETWEEN BA AND AC IN CANADA IN EACH
DIRECTION
1 FREE BETWEEN AA/BA/US AND WS IN CANADA IN EACH
DIRECTION
1 FREE BETWEEN AA/AY/BA/IB/US AND HA IN HAWAII
IN EACH DIRECTION
2 FREE BETWEEN AA/AY/BA/IB/US AND AA/AY/BA/IB IN
NORTH AMERICA IN EACH DIRECTION
FARE BREAK SURFACE SECTORS NOT PERMITTED AND
EMBEDDED SURFACE SECTORS PERMITTED ON THE FARE
COMPONENT.