Air carrier First Officer reported ATC turned them on to final inside the final approach fix resulting in an unstable approach and low altitude alert from Tower.

Narrative

I was the PF; and the CA was PM for this flight. We were in the final descent and approach on a downwind leg into ZZZ; when ATC gave us a base turn inside the FAF (ZZZZZ) on the ILSXX approach. The CA then stated that we had both the preceding traffic and the airport in sight. The approach controller issued us a Visual Approach clearance to runway XX. We were much faster and higher than I would have liked to have been; given our proximity to the FAF and the airport. ZZZZZ is charted at 1900' MSL on the approach plate. Since we were inside of that; and I had the field in the sight; I bugged 1000' in the Alt Select window. In an effort to get down; I selected LNAV which made the base to final smart-turn". FLCH was then selected; and Approach mode selected; armed in standby. I was anticipating the aircraft would keep descending in FLCH until it caught the LOC and GS; but that it would happen prior to reaching 1000'. While getting flaps and gear in; the airplane descended quicker than I thought it would; leading to us leveling off at 1000' MSL; much lower than the GS. This prompted the tower controller to issue us a low-altitude alert. Both the CA and I stated that we had the field in sight and were correcting. When I looked up; I saw 4 Red lights on the Runway XX PAPI. I asked the Captain if he wanted me to go around or climb. He said no we're fine; as we were slowly starting to pick up white lights on the PAPI; getting back on the proper descent path/angle. We captured the LOC and GS shortly after and landed and taxied to the gate without further incident. At no point in time did we receive any GPWS alerts. After debriefing and thinking more about this flight on my own; I realized that in the hustle of getting configured and trying to get down; I incorrectly selected 1000' in the altitude select window instead of 1100'. ZZZ Runway XX TDZE (Touchdown Zone Elevation) is charted at XX'; so rounded up; and adding 1000' would have been 1100'. I cannot recall if we were fully configured prior to reaching that 1000' above the TDZE altitude; required for our stabilized approach criteria; hence the report. In hindsight; realizing the "bad" vector; we should not have accepted a Visual Approach; and instead been vectored around for a more manageable and stable approach. In this specific instance; I had the airport in sight the entire time - instead of fumbling around with the FGP (Flight Guidance Panel); I should have just disconnected the AP (Autopilot); and hand flown the remainder of the approach."