General aviation pilot reported descending below a charted altitude on an approach outside the FAF in mountainous terrain while IMC; and received an altitude alert from ATC. The pilot climbed then returned for another approach.

Narrative

I flew into Aspen (ASE) and deviated from the published instrument approach procedure (LOC DME E). I was told by the approach controller to expect vectors to Jargu; an IF on the approach. I loaded the approach and; upon being given an initial vector; activated vectors to final on the FMS. However; I noted that the Minimum Enroute Altitudes (MEAs) for all but the Final Approach Fix (FAF) were not showing next to the fixes in the FMS. I had my chart on my iPad; which I used to verify the MEAs. I was given a last vector to join the LOC and was cleared for the approach with the restriction to cross Jargu at 13;000 ft. I turned to my assigned vector using heading mode automation and hit the approach mode on standby. The LOC was captured; and I was established on the approach at 13;000 ft. My plan was to manually set the step-down altitudes and VS down to the subsequent fixes in the approach. Once established; my FMS was indicating that my next fix was the FAF; DOYPE; which has an MEA of 11;700 ft. I set my altitude bug to 11;700 ft and began my descent using VS. In reality; I was crossing on the KICER to FIMSO leg which MEA is 12300. At this point; I was still in IMC and had no reason to believe I was deviating from the procedure. As I crossed approximately 12;300 ft and descending; I was asked by the approach controller if I had the field in sight. I replied 'negative.' He then gave me a minimum altitude alert and instructed me to climb immediately to 13;000 ft. During the interaction; I had descended another 400 ft before arresting the descent and beginning to climb. I was then cancelled of my approach clearance and vectored back around for another attempt.I take full responsibility for the deviation. In hindsight; I relied too heavily on the FMS and didn't adequately cross-check my position and altitude using DME. I've flown into Aspen many times; which may have contributed to complacency on my part. I relied too heavily on the FMS. If I would've been prepared and crosschecked with DME to identify the fixes; the deviation would have not occurred. I've scheduled instrument flight training to fly this particular approach procedure and figure out why the FMS was not sequencing the fixes. This incident was a valuable learning experience. I appreciate the approach controller's assistance during the second approach; by giving me position checks; which helped ensure a safe and successful landing. The most important lesson is to be prepared to crosscheck with DME on all localizer approaches and not rely solely on FMS.To improve safety; I recommend considering the use of an Initial Approach Fix for clearance to the LOC DME E approach in IMC. This might help reduce pilot workload and minimize the risk of deviations in a challenging approach like ASE.I'm committed to safe flying practices and will take all necessary steps to prevent similar incidents in the future.