This is correct, I went down the rabbit hole a few years ago on this unusual unit. Here’s another of their ambulances from that same day, note the same livery and siren.That was a Long Island College Hospital Isuzu/Braun unit, from Brooklyn Heights.
It’s a SP (Scientific Prototypes) Mark 7, like OP said. Good chance it’s a former NYPD unit too, a lot of private ambulance companies used them into the early 00’s. Most notably Columbia Presbyterian, there’s a few clips of them on YouTubeI'd like to know which siren and air-horn it was using!