Captain reported multiple low altitude alerts from ATC on approach; went missed approach and diverted to an alternate airport.

Narrative

This was a Medical Donor Flight with Four Medical personnel on board. We first received notification of the flight in the mid-afternoon. The co-pilot had retired to bed early to prepare for a XA00 departure for a part 91 reposition flight. There was some confusion as to the desired departure time out of ZZZ [Airport]; so the captain was not able to retire to be in preparation of the flight until around XV00 and thus did not get much sleep. During the attempted rest; the departure time was moved earlier by an hour. The crew showed at the airport at XY30. There was thick frost and ice on the aircraft. 70 gallons de-ice was applied to the aircraft before the departure. The airframe was clear of ice when we departed. The flight to ZZZ [Airport] was uneventful. We then flew to ZZZ1 [Airport] and arrived there at around XC45 without any issues. We had reserved a sleep room for the crew at the FBO. Co-Pilot was able to get another couple of hours sleep. The captain was able to get about 1 to 1 1/2 hours sleep. The Medical staff notified the Co-pilot that they were 90 minutes out then again at 30 minutes out they said they were returning to the airport at which time the co-pilot awakened the captain. Our arrival was changed from ZZZZZ3 to the ZZZZZ2 prior to ZZZZZ1 [Intersection]. In the descent there was a lot of mountain wave and moderate turbulence which was causing the Autopilot to disconnect multiple times. At XI02 when ATC vectored us off the ZZZZZ3 arrival. We were prepared to do the ILS XXR approach. The winds were very gusty in flight and there was some concern that the APCH (Approach) might not be able to be done because of excess tailwind component. If I remember correctly the winds were 060@15. I checked the ATIS multiple times and then heard that they were using RNAV YYL approach. Air traffic control changed our approach to the RNAV Y YYL approach while we were being vectored for final. I had already put RNAV YYL approach in the FMS but was frantically trying to find the other APCH which we did not have in our database. I found the approach on my iPhone and it was an RNP 2 approach which we were not approved for. While I was busy trying to find and program the changed approach into the box; I told the SIC (Second in Command) to just fly the original RNAV YYL approach. ATC gave us a short vector to final and asked if that was too tight. I responded that it was perfect. Which at the time I felt was appropriate. We observed the proceeding aircraft call a missed approach. With the autopilot kicking off many times during the approach; the flying pilot had his hands full. My attention being diverted to trying to find the other approach; to no avail. While on final approach we received 2-3 'Low Altitude Alerts' from ATC. Each time we made altitude corrections. We ended up going missed approach and going to ZZZ2 [Airport] as our alternate airport. We were being vectored for ILS ZZZ in ZZZ2. On about late downwind ATC asked which runway we wanted. We told them we would like the RNAV YYL at ZZZ3 [Airport]. While the weather was more conducive to getting in at ZZZ3. I was on the number 2 com trying to contact the FBO to get them to communicate our divert to the ground transport team. The turbulence was still kicking off the autopilot. We were vectored through final then given a heading of 090 to intercept final. We intercepted final approach and again the auto pilot kicked off. Again; we received a 'Low Altitude Alert' I believe around ZZZZZ2 [Intersection] or ZZZZZ3 [Intersection] from ATC. I prompted my co-pilot to follow the vertical guidance which by this time he was so rattled and beat up that that he really was sluggish in responding. We then had the field in sight and proceeded to a very smooth landing.I attribute these events to FATIGUE aggravated by auto-pilot malfunctions; weather; turbulence; delayed vectoring for both approaches; and the captain's attention diversions due to an improperly assigned approach and the need to get the donated organs into the operating Room. This was a classic accident chain that was thankfully broken by the observant approach controllers that called 'low altitude alert' several times during the first approach.