Air carrier Captain reported they descended below the published altitude of an RNP Approach and ATC issued a low altitude alert.

Narrative

I was pilot flying into DCA. We were originally set up for the LDA 19; as advertised on the ATIS. We briefed the approach in its entirety. At some point with approach control; we were given the RNP Z 19 and programed the flight computer accordingly. After our debrief; we agreed we did not do a thorough brief of the flight computer for the RNP approach. I accept responsibility for this; as I have flown into DCA other times this previous week. Complacency got me and I learned a valuable lesson. As we were navigating the Freedom arrival; we were told to cross Darci at above 2;700 and clear for the RNP Z 19. I armed the approach and final approach engaged. At some point on the approach; the FO noticed the airplane was not tracking the altitudes; and I elected to turn off the automation and attempt to hand fly. Most circumstances this would be appropriate; however after a debrief; I don't feel this was the best decision We were extremely high and I attempted to get back on the PAPI's and around 500 feet; we decided this was not going to work and executed the go around. The go around went textbook.We were given vectors to do another approach. We elected the RNP Z 19 once again to see if it would work; however; in hindsight; we should've just done the LDA/Visual. While on vectors; ATC advised that we were a little bit close to the prohibited area and to use caution this next time around. I even called the tower on the ground to discuss how this situation played out; and he said there was no further issues with the prohibited airspace. All was good.On the second approach; we were cleared for the approach and noticed that the altitudes did not load in the flight computer. This time; I also elected to hand fly and follow the altitudes on the chart while flying along the river. I ended up about 100 feet low; I believe; at one of the altitudes and got a low altitude warning from ATC which I immediately corrected. We continued on and landed uneventfully. I still don't understand why the altitudes for the RNP didn't load correctly; or at all for that matter on the second approach. Operator error? Definitely possible; I just don't know. Once we got off the airplane; I elected to call the local ATC guys in the tower and apologize and talk about our situation. They said they reviewed our distance from the prohibited area and decided we did not penetrate or violate anything. And actually said they were rooting for us on the first approach and they thought we were going to be able to get on the ground. We had a good discussion.I then called Check Pilot friend of mine to further debrief the situation to see what went wrong. Talking to him I learned of an aircraft type anomaly from that arrival into the approach; which I think is exactly what happened to us. Since we did not properly look at the flight plan page; we never saw a discon and airplane went into final approach. Everything looked normal to us as far as Flight Mode Annunciators. However; after a lengthy debrief; we remember we were laterally following the arrival course. We were given an approach clearance for 19; but we never realized the box wasn't going to sync up; because the arrival and approach don't connect. Again; this is our fault for not identifying this earlier. Although to be honest; the lateral track off the end of arrival and the RNP approach was identical and so close together; I never caught it. Which explains why the airplane didn't actually descend even with FINAL APP armed. I think another contributing factor was ATC left us on the arrival and then gave us the approach clearance. And we mistakenly assumed it was linked up; because it sure looked like it was initially. But that arrival doesn't connect to the RNP approach; I know this now.