I’m a daily commuter through RCP, but going against the commuting flow (north in the morning, south in the evening). The traffic on the road really depends on the time of your commute. Prior to 6:45 am, the worst backup is where Beach Drive meets Rock Creek Parkway. The stop and go traffic can extend beyond the zoo entrance. By about 7:15, expect heavy stop and go traffic between Broad Branch and Piney Branch due to the red light at Park and the stop sign at Piney Branch. After Piney Branch, it flows pretty smoothly down to the parkway, but expect lots of cars.
If you want to take the trail, you can take Bladgen and jog north for a bit to catch the trail at Broad Branch. If you take Piney Branch, you can go south to Klingle Road and get on the trail there.