My only experience here is as an SLF who flew frmo MAN - LHR one evening on an A319 with one passenger (me) four cabin crew, one deadheading pilot and two pilots. as we taxied out I saw two A330 ME aircraft taxying to the end of the runway and I hoped we would by-pass them; which we did. We took off from a long way down 23R (estimate less than 1,900m remaining) and were airborne very quickly.
Whilst I appreciate that this is a different scenario, in terms of pure physics, a single engine take-off of this 737 on a 4,000m runway should be possible, even with a gentle run up. if the flight was only a couple of hundred miles to an easier airport to fix the second engine... I bow to the experts when it comes the operational side of matters which render this idea not worthy of consideration.