Cleared for a Mt. Vernon Visual Approach to Runway 33 at DCA the flight crew of a CRJ-700 mistook Runway 1 for their cleared Runway 33 due to bright lights on the former and dim lights on the latter. I missed approach and excessive banking to avoid Prohibited Areas resulted.

Narrative

During Operating Experience at night into DCA we received the Mt. Vernon Visual Approach to Runway 33. During the approach Runway 1 was lit up for landing. We could not see Runway 33. At or about 1;000 AGL the First Officer mentioned that we were lined up on Runway 1. I concurred and move back out over the river to find Runway 33. Fully configured we noticed Runway 33 passing on the left (not very well lit up). We were down to around 500 FT and I realized we could not land; the runway passed by on our left; due to not being lit very well. I turned toward the runway; receiving a 'bank angle' warning and executed a missed approach; this turn was to stay south west of the Potomac River and not violate P-56. After the go around we were vectored back around to the same Mt. Vernon Visual and cleared to land on Runway 1 which resulted in an uneventful landing.The threat here is a difficult visual approach made harder by a dimly lit runway overpowered by a brightly lit runway that we were not cleared to land on; the lit runway really pulls you to it to land. Considering the complexity of the airport and the threat of P-56 we were quick to execute the go around. If we are cleared to circle to 33 we need to ensure the 33 is lit up sufficiently to see during the maneuver. Not having done this before the runway really comes up fast. It just needs to be lit up better. [It was] my fault to not ask if it was lit up.