Air carrier Captain reported receiving a low altitude alert from ATC due to descending below profile while on an RNAV RNP approach. Flight crew discovered post-flight the aircraft was not authorized for RNAV RNP approaches.

Narrative

We completed our previous flight and before swapping to another aircraft for our subsequent flight to DCA airport; I looked over the flight plan for the next flight before running to a nearby vendor to buy a breakfast sandwich. We had about 01:15 between flights. When I arrived at our next aircraft; the F.O. had already loaded the flight plan and offered to do the walk around while I ate and reviewed the flight planning and performance. I noted that DCA was conducting landings to the south on runway 19 and that the F.O. had loaded the RNAV (RNP) Z Rwy 19 for the approach into DCA. The flight to DCA was uneventful until we began the approach and realized we weren't getting good descent guidance. We could see we were going to be high on the approach and consequently asked for and was given an early missed approach. We were cleared back around and not sure for the reason of the inaccurate guidance on the previous attempt; hard-tuned the DCA DME and briefed that we would use the DME distance as back-up for our initial descent elevation information (3 to 1) and then transition to the runway VASIs (Conditions were VFR throughout the arrival; approach; and landing). We subsequently experienced the same invalid descent guidance but using our back up plan; flew a more normal descent. While bracketing our altitudes to correspond with the DME readouts; at one point we got below our profile prompting ATC to give us a low altitude alert which we promptly corrected for. Using the runway VASIs thereafter; the remainder of the approach and landing was uneventful.On the ground after debriefing our ordeal; it occurred to me and looking again at our flight plan; confirmed it - that our airplane was not authorized for RNAV RNP approaches. I recalled reading the MEL when I reviewed the flight papers but failed to apply the restriction later when we arrived at our new airplane. Pointing this out to the F.O.; they also had a 'ugh' moment; explaining that they also previously noted the restriction. It was obvious now that we missed incorporating the restriction into our flight operation.