Air carrier flight crew reported flying an unstabilized approach on a charted visual approach and elected to continue to land rather than go around.

Reporter 1

During the approach phase of flight to the river visual approach to [Runway] 19 at DCA, ATC issued a vector to intercept the approach in the vicinity of the GRAYZ intersection on the LNAV approach overlay. We had briefed the approach to include landing data and runway exit strategy and agreed that a flaps 40 setting would be the most effective. This leg was the third and final flight leg of the day. Prior flight legs had included significant weather mitigation and planning on our part as a crew, and I was, by this time, moderately fatigued. The altitude was approximately 2,000 [ft.] MSL, as I remember, which was issued to us by ATC. The aircraft was being operated using the autopilot, auto- throttles, and ATC had slowed us to 160 KTS for spacing. Because of the issued speed, the aircraft was configured with the landing gear down, flaps 15 for the speed, and subsequent approach. As we began to approach the intercept point, I realized that the intercept angle to capture was going to cause an overshoot, a subsequent overfly of the river to the opposite side, and in proximity of the restricted airspace. I dis-engaged the autopilot and manually flew a course to intercept the LNAV overlay of the river, ensuring no overflight of restrictive airspace. Once on the LNAV course and in the center of the river, I reached over and re-engaged the autopilot, or thus I thought I had. My hands were on the control column and when I selected the autopilot, it felt as the autopilot had engaged. The Pilot Monitoring did not alert me to the fact that the autopilot had not engaged. I was feeling time-compressed to get all the required tasks completed before landing and wanted the flight path to continue a descent using VNAV on the arrival. I reached up and set a lower altitude on the MCP panel. I felt the aircraft start a left descending turn, which I expected as we were following a left turn in the river. As the pitch attitude of the aircraft decreased, we received a GPWS Obstacle Alert (probably the Washington Monument in the distance). I immediately responded by manually correcting the flight path and then realized that automation had not been activated. I stabilized the flight profile and began using the river terrain features and requested flaps 30 and Landing Checklist. The aircraft was still in the center of the river, but now not on vertical profile slightly high. The time and approach compression was significant. As I began to turn final and line up on the runway, approximately 800 [ft.] MSL, the Pilot Monitoring queried me if I wanted flaps 40. I said yes, and flaps 40 was selected. I was above the PAPI and used normal maneuvering to re- align myself to a normal landing profile. I landed well within calculated Touchdown Zone and taxied to the gate. At no time were any aircraft limitations exceeded nor any airspace vertically or laterally exceeded, nor ATC clearances exceeded.

Reporter 2

Captain was PF (Pilot Flying) into DCA on the River Visual Runway 19. ATC gave us a heading 130 to fly and then clearance for the River Visual Runway 19. With the heading we were given, we agreed to go direct GREYZ. Because of that heading, 130, it was evident that the autopilot probably was not going to anticipate the turn enough, leading to a possible overshoot of the runway and getting close to the Prohibited Area, P56B. The Captain chose to disconnect the autopilot to lead the turn. He then tried to reengage the autopilot, while I thought he was still hand flying. The autopilot did not actually engage, and our descent rate increased. Right as I was about to call out, "sink rate," the GPWS called out, "sink rate," instead. The Captain realized the autopilot was not engaged and corrected the flight path. At that point, we were descending at about 2,000 FPM and about 1,500 [ft.] AGL. He recovered nicely, However, it put us below stabilized criteria and we selected flaps 40 below 1,000 [ft.] AGL. We should have gone around due to the unstabilized approach. We ended up running the Before Landing Checklist late as well. We landed normally and continued to the gate. It was a quick set of events, and we were both busy getting the aircraft back where we wanted it, but ultimately, we should have gone around.