A pilot circling to land from an ILS approach reported they deviated from their intended course and received a Low Altitude Alert from ATC.

Narrative

During a night VMC arrival into TEB we set up for an ILS Z Runway 06 with an initial approach fix at VINGS intersection. The last ATIS we received showed favorable weather with the possibility for LAHSO operations on both runways. As a crew decided we would not accept LAHSO due to limited distances. The Approach Controller issued the standard Runway 06 ILS clearance when we were approximately 2 miles west of VINGS; with the addition of 'expect to circle to land Runway 01'. This was the first time we heard this requirement; and the approach was duly read back/accepted.Once passing VINGS at 2000'; there are multiple crossing fixes at waypoints LEESY (1700A) and a mandatory 1500' at Dandy; after which descent to 1;300 is allowed prior to reaching waypoint TORBY; from which Teterboro Tower directed to commence the circle to Runway 01.Due to the requirements to change configurations and the multiple crossing constraints; any change to the FMS was delayed until we had passed DANDY and were descending to 1300'. During this intermediate approach segment the aircraft was configured for the circle to land with Gear Down / Flaps Full; and a manual speed of Vref+10 for the circle to Runway 01. During the level segment to TORBY; while still on the extended centerline of Runway 06; I instructed my First Officer to insert a 4.0 NM final approach to Runway 01 so that we had valid landing data and an electronic 3.0 degree path to the runway. Normally two VP waypoints (VPEZA and VPDAU) are used to track from TORBY to a short final approach to Runway 01; but due to the lack of advanced notice of the circle; the Runway 01 visual approach was our best choice to limit 'heads down' time at low altitude. Prior to activating the FMS change; I selected Heading Mode 060 degrees on the Autopilot in anticipation of losing the waypoints from the ILS; including TORBY. Based on my last reference to my Navigation Display; we were just about to cross overhead TORBY (and we definitely did not commence the circle from DANDY). The First Officer complied with this instruction; but due to being at low altitude I failed to validate his inputs to the 'Final Approach Distance' dialog box on the FMS. (More on this later.) Seeking to reduce workload and due to night/dark conditions I asked for navigation direct to the 4.0NM final fix and requested the altitude selector be set to 'Zero' so that the VNAV function of our Autopilot could properly calculate a path to the runway. Based on my previous study of this maneuver (which has a history of causing issues in its execution) I knew that a 4.0 mile final to Runway 01 would keep us just outside (north) of Newark's Class B airspace while allowing me time to gain visual contact with Runway 01 and it's PAPI installation. It was during this segment that the TEB Tower Controller issued an airspace alert and directed us to fly toward the TEB airport as we were about to enter Newark's Airspace. In response I disconnected the Autopilot and turned east northeast towards the MetLife Arena complex and the extended Runway 01 centerline. The disconnection of the Autopilot and turn towards the airport resulted in the controller later issuing a 'low altitude' alert which we acknowledged. I immediately maintained our altitude which I recall to be approximately 950 MSL.Shortly afterwards we were aligned with Runway 01 final and picked up the PAPI to execute a visual landing. After exiting the runway we taxied into the ramp without further issue. We then set to our task of securing the aircraft and offloading baggage and perishables.During the shutdown process I went over the 'flight log' portion of our FMS to better understand just how this situation could have taken place. It was at that time that I noticed the FMS distance between the Final Fix and Runway 01 was exactly 5 Miles; not 4. This data was also still on the Route page of our displays as the FMS did not sequence because we never overflew the 5.0 NM fix. While seemingly insignificant; this resulted in a more southerly course from over TORBY than would have been the case if the waypoint was a mile closer (and further north of EWR's Class B).The following are the 'lessons learned' from this experience.1. Unbriefed procedure. The circle to land maneuver was a 'surprise' and we did not have adequate time to prepare as normal for this maneuver. The ATIS we received did not mention circling as a possibility. Further; it was a last minute change from Approach advising us of this requirement. Nonetheless; it was at this point time to say UNABLE. Perhaps due to familiarity with the airport and its environment; or perhaps due to 'mission-itis'; as Aircraft Commander I accepted a challenging clearance that had not been properly briefed or anticipated. It was entirely my responsibility to say 'NO thanks' and seek other options such as another runway; or a diversion.2. CRM. I did not properly supervise the input of data to the FMS to confirm the First Officer's changing the final approach distance to 4.0 NM; versus the default value of 5.0 NM. This is 'Automation 101' flight discipline. It is something that while seemingly small; set off a chain of stressful events that impacted our situational awareness and resulted in everything that has followed including this report. As the Pilot Flying; it is a given requirement to always CONFIRM FMS inputs from the other pilot; even; and perhaps especially; while at low altitude.3. Fatigue. This flight occurred at the end of a 15.5 hour duty day. We thought we had mitigated this fatigue through inflight rest; and briefing that we would fly a stable approach while making maximum use of available automation. What we failed to discuss; and what is an excellent lesson; is the insidious nature of fatigue and get-home-itis that lays in wait for those who do not continuously update their mental model; and proactively seek information; or options; once a situation has changed from benign(straight in ILS 06) to challenging (circle to land Runway 01).4. Reflection and Learning. My FO (First Officer) and I debriefed this event after we had shut down the aircraft. I emphasized to him that although we experienced this as a 'crew'; it was ultimately my duty as PIC (Pilot in Command) to be the 'final authority and ultimately responsible' for the conduct of our flight. Allowing items 1-3 to occur negatively impacted flight discipline. This was fully my responsibility; and comes with the mantle of command. It is now my duty to learn from this experience going forward.5. Gratitude. I am thankful that the system works as required when things do not go to plan; and that ATC very quickly got involved so we could get safely on the ground. This report is a reflection of my commitment to continuous improvement; and is an honest self-critique. I am sincerely 'GLAD TO BE HERE' to write this report; and to learn from this experience.