Yeah there's also space on May 14th to go JFK - GRU - EZE on TAM 777 in J.

OP, if you don't have a BA account sign up for one and search on there. You're going to have to put some of your own time and effort into this search but especially on the T-331 date there are options available if you're creative, do the legwork, and start by finding the big route and then figure out the DCA - XXX portion.

Just book as two one-ways. Make a one-way booking that leaves on T-331 (say today) and then check again in 2 weeks on the day you want to return or thereabouts and see if space has opened up back the other way and book a one-way return. AFAIK there's no benefit on AA for booking a RT so that's a good way to grab space as it becomes available at that T-331 point