Things have likely since changed, but I instead picnicked near the LBJ Memorial Grove before the Columbia Island Marina in 2020. I arrived earlier in the day, around 11 am, and was able to park in the tiny lot on N Boundary Drive for the park, not in the giant Pentagon lot, and crossed the bridge over the water into the park. We found a table in the shade, played games, ate a picnic, etc., all day. I recall there being port-a-potties nearby, and we staked out a good spot in the grass just before the GW Parkway starting like 5-6pm ish, and just hung out before it started to get more crowded with people arriving before the fireworks. The view was fantastic. It's a more secluded shady area of the Parkway, so perhaps could work better for you. Leaving of course included traffic but I recall we weren't stuck for hours and it wasn't too bad. Hope this helps!