Fractional Captain reported receiving a low altitude alert from ATC during a confusing and high workload period. The crew did not understand which Localizer approach ATC expected.

Narrative

On a [flight] from ZZZ to ASE; I (PIC/PM); was talking with ATC and setting up the FMS with navigating procedures. In route; the PF and I spoke about newly assigned LOYDD 1 arrival and the weather in Aspen. Checking the digital weather in Aspen it said the weather was 10 miles and clear; smoke and haze. The PF said it's clear; let's plan a visual approach. I said OK; let's back it up with the RNAV (GPS) F. I loaded the approach in the FMS; and the PF gave a complete overview briefing. We talked about possible go-around scenarios anywhere along the descent in relation to terrain; wind shear; short final; or a balked landing. We discussed trying to be fully configured for landing by Red Table. Getting close into the Aspen area; I could start to pick up the ATIS; which was difficult as it was coming in broken and intermittent; finally received; I briefed the PF. DENVER Approach advised plan [Runway] 15; vectors for the LOC DME. I began doing the Top of descent checklist and changing the FMS for the new approach. I saw the LOC 15 approach in the FMS and selected if via AJAXX. The PF gave a quick briefing from the LOC DME E approach plate. We got vectors for the approach and to join localizer. We getting busy completing the approach checklist; configuring the plane; watching terrain; looking for the airport. We could see the road leading into the Aspen airport. Received the clearance for the approach; an ATC hand off to Tower. Approaching abeam RED TABLE; PF called for gear down and landing flaps; and he started the descent. And called for before landing check list. I could see the terrain below us; checked in with Tower; they asked if we see the airport; I replied not yet; we see road leading in. Tower then advised; low altitude alert; I checked altitude it showed 12600 and descending; I looked for our position on the chart; could still see the terrain. Tower advised us again about low altitude alert; now at 12300 and descending; advised 12600 for our area; I immediately took control of the aircraft; disconnected the auto pilot; added power and climbed back to 12600. We called the airport in sight. Tower advised we were too high and close in for a safe approach; and gave vectors for reroute back for another approach. I continued as PF. I had noticed the fixes on the MFD (Multi-function Flight Display) were different than on my approach plate. I had the PM renter the approach; as we knew we weren't getting a visual on the airport until lower and close in; that's when we noticed there was a LOC 15 and LOCDME. We loaded the LOC DME E AP (Approach) and checked each waypoint and altitude. New briefing and checklist completed; we got vectored for the approach. I gave the controls back the other pilot. Denver was busy; they gave us a heading to join the localizer and switched us to Tower. He was busy as well; I advised we needed our approach clearance; he asked; a visual; I replied; we need the localizer clearance. We were at 12900 about over FISMO. He cleared us; we descended and called airport in sight. He issued a cleared for the visual approach. We got on profile and landed normal.On the ground we discussed what happened. We were confused why there were two approaches to the Runway 15; both a LOC. We discussed the importance of verifying the waypoints in the FMS to the approach chart; even when we think it's VFR. We discussed; when ATC advised about an altitude alert: react; climb or level off; don't continue descending; they may see something we don't. After I got to the hotel; I started researching the approaches; wondering if it was an old a ILS 15; why is it in the FMS; why don't we have 2 approach plates? Why wasn't ATC being very specific about the approach they were clearing us for? And stating the LOC DME E. Upon researching the approaches. I found information related to the topic. A website has a topic of discussion about the Aspen approaches. Within the article I found a link to an FAA Letter to Airmen highlighting a common error of loading the LOC 15 rather than the LOC DME E. I contacted the Safety department of my company; and briefed him on what had happened and what I had found. I forwarded him the Letter to Airmen. He forwarded it to the Chief Pilot and the Supervisor. They issued an urgent company bulletin and mandatory reading of the Aspen letter.Perhaps; 'Special' approaches that require an LOA should only be available through an add on for the FMS. Or be label in the FMS as SPEC ILS 15; to prevent it from being used.