Air carrier Check Airman reported receiving a low altitude alert from tower during a night visual approach.

Narrative

We were cleared for a visual approach to runway XXR approximately 5 miles northwest of the airport. We continued on a left downwind arrival utilizing an RNAV (RNP) approach as a backup. The student who was Pilot Flying (PF) began a descent to the FAF altitude based on an RNP approach of 1100 feet AGL and at a rate greater than 1500 FPM in Flight Level Change (FLCH); but were still about 5-6 miles from the airport. This prompted ZZZ Tower to issue a 'Low Altitude Alert' at which time I regained controls; overrode the automation and climbed to a higher altitude. I had the airport continually in sight and was positioned to safely land the aircraft without further incident.We continued the flight too long in heading mode matched with a lower altitude that was entered below the MVA in the area. The students use of FLCH; an extended downwind; a high rate of descent below 2500 feet AGL; and my late decision to discontinue this approach and regain controls sooner helped lead to this event.My idea and expectancy were that the student had been showing improvement in performing visual approaches and had accumulated over sixty hours of training time. Therefore; it seemed prudent to let them continue this approach with minimal guidance in order to build their confidence and proficiency. However; I should have overridden the automation sooner and/or reset the altitude alerter to a higher altitude to ensure that we kept a 300 feet per nautical mile altitude from the airport at all times. A go-around would also have been wise; but given the student's current state of being overwhelmed and their limited ability to accept commands; I felt a safe landing from our present position was a better course to take to safely complete the visual approach to a landing.