A319 Captain reported they did not comply with altitude restrictions on an approach to an airport in mountainous terrain due to misconfiguring the automation despite warnings from the First Officer.

Narrative

I was CA and PF on the next-to-last leg of a four day sequence. ZZZ is a Special Qualification Airport; due to mountainous terrain and complex procedures; requiring pilot review of the company ops advisory pages and Jeppesen color qualification pages in Jeppesen FD-Pro prior to operations there. The FO/PM had flown into this airport numerous times at a previous carrier prior to joining our company; so is very familiar with the local terrain and procedures. This was my first time flying there; however.Enroute to ZZZ; I reviewed the ops advisory pages and the color qualification pages; and we discussed the ILS Rwy XX approach planned based upon the prevailing winds at the time; along with the company engine failure missed approach procedures for that runway. Once in radio range of ZZZ; the FO/PM obtained the ATIS; which reflected a new weather observation that included a wind shift and runway change to runway YY. I then adjusted the FMS to reflect the ILS Rwy YY approach and we discussed the new approach. The FO warned me that their previous carrier had had a terrain warning event on this approach; where pilots that were cleared for the approach had immediately set the FAF altitude at ZZZZZ [Intersection] of 3;800 MSL and begun a descent to that altitude prior to capturing the glide slope; failing to honor the stepdown altitudes of 5500 ft. at ZZZZZ1 [Intersection] and 4800 ft. at ZZZZZ2 [Intersection]. As a result; this approach became a regular focus during their recurrent simulator training. I noted that the approach does not allow autopilot coupled approaches; a constraint I have not seen before; and said that I would hand fly the approach and use manual thrust for proficiency which I try to do at least once on every sequence. The FO/PM suggested we could fly the GPS approach that allows coupled approaches; but I did not consider that to be necessary. In hindsight; that would have been a better choice than the ILS.I failed to recognize the threat the FO was warning me of; which is especially salient in the Airbus; that once the localizer is captured the aircraft will NOT honor FMS (NAV) altitude constraints. My plan was to not arm approach until inside of ZZZZZ2 [Intersection]; and instead use a managed descent; allowing the FMS to ensure the ZZZZZ1 [Intersection] and ZZZZZ2 [Intersection] constraints were met. I then failed to properly monitor the aircraft to ensure it did what I intended. Once we were cleared to intercept the localizer and it captured; I set the ZZZZZ [Intersection] FAF altitude of 3800 ft.; and managed the descent; while we were still outside of ZZZZZ1 [Intersection]. I had forgotten that; once in localizer track mode; NAV altitude constraints will not be honored. The aircraft was slow and smooth to start the managed descent; and I didn't recognize that it had started descending. The FO/PM did; however; and warned me. As I recall; I then mentioned my intent to have the FMS control the descents to avoid being below the ZZZZZ1 [Intersection] and ZZZZZ [Intersection] fix altitudes in case the glide slope did not honor them. The glide slope feather does not extend beyond the FAF on this approach; indicating it is not certified beyond the FAF. The FO immediately pressed ALT HOLD and pointed out we were below the required altitude at ZZZZZ1 [Intersection]. Reviewing the incident; we passed ZZZZZ1 [Intersection] at 5325 ft.; 175 ft. below the minimum of 5500 ft. We passed abeam the highest peak on the approach plate and very close to the final approach course at 5125 ft. (minimum altitude at that point was 4800 ft.). As I recall; I then armed the approach mode and we captured the glideslope; passing the next fix of ZZZZZ2 [Intersection] at 4875 ft. (minimum altitude of 4800 ft.). I then turned off the autopilot per the approach requirements; and autothrust per my plan. We passed the ZZZZZ [Intersection] FAF at 3850 ft. (minimum altitude and crossing altitude of 3800) and continued to landing.Fundamentally; I failed to properly monitor the aircraft to ensure it actually flew the vertical path I intended; and intervening when it did not. Factors that led me to misconfigure the automation include several approaches commonly flown require stepdown fixes prior to the FAF. However; those approaches are normally flown in NAV; managed descent; until approaching the FAF; when approach is armed. Those set me up with an expectation bias of being able to manage the descent; forgetting that on this approach I was in LOC; not NAV. An earlier design (since reconfigured) of the ZZZ1 ILS ZZ approach was reported to have a number of pilots violated for crossing a fix outside of the FAF below the minimum altitude because they armed approach; captured the GS; and it brought the aircraft to an altitude below the minimum allowed crossing that pre-FAF fix - that made me leery of arming approach prior to the FAF.This is a challenging destination for our Airbus pilots. Include this airport as a focus in recurrent simulator training. Discuss the challenges of the mountainous terrain; coupled with a hand-flown approach; and stepdown fixes prior to the FAF yet outside the GS service volume. Add a photo from the flight deck on a clear day of the terrain ahead when over ZZZZZ1 [Intersection].