I doubt that the fare would go lower, as your departure date is less than a month away. Most "discount" business class fares target the discretionary leisure traveler by requiring a 50+ day advance purchase. $5.5k sounds reasonable for a business class ticket purchased this close to departure.
I'm not familiar with AM J, but I know not all of their seats are fully flat yet. Connecting via GRU is a pain. I highly suggest just landing in GIG. I, too, live near DCA. Last December, I flew to GIG on DL and liked the timing and the J service.