Frankly, you'd probably be better off on the QF service to JFK. From there, you can fly to WAS on AA/DL or take an onward flight from LGA if you want more options. Depending on your final destination, it might be faster to take Amtrak from NYC to WAS.
Have you checked other routes? Specifically, routes that don't use LAX as the gateway. LAX lacks frequencies for many routes, making it an annoying gateway for the East Coast (as you've seen). The DCA perimeter rule isn't doing you any favors either.
Finally, have you looked at MEL-NRT-DFW-DCA or MEL-DOH-IAD?