That looks like it's got a few miles on the clock, but - if you'll forgive the impertinent question - do you know that the backlash is because of the screw / nut? Have you checked the mounting of the nut to the bed?, and have you checked the mounting of the screw and the coupling to the stepper motor? if the problem is elsewhere than the nut you could save yourself a lot of hassle.

If you're confident it is the screw/nut - then from what I've read in the last 10 minutes some of the 3040/trapezoidal screws use a delrin (plastic) nut - and that (rather than the screw) is likely to be the cause of the backlash - although 1mm is pretty horrendous - surprised the nut is still holding together with that. I'd expect if it is the nut that it's got progressively worse to the position that you find yourself in today. If it is delrin then it's not hard to re-cut a new nut using a bit of heat, delrin, and the original screw. Otherwise...

Eyeballing the photos you've provided - it looks as though the nut is attached to the carriage with two M4 or M5 bolts - use a hex key/drive to unscrew those and you should be able to persuade the nut from the carriage - then you can properly assess it for backlash. I've spent a short time googling/ebaying/amazoning and the T 14 4mm p2 trapezoidals are not immediately obvious from the "usual" sources. It might be worth looking to see if you can get similar, even if not the same. Maybe the Chinese suppliers (including Jazz's recommendation) may be able to source a nut.

I'm purposely ignoring for now your reluctance to perform a lot of disassembly/reassembly - that might be unavoidable.