A couple of techniques worth considering:-
One is to turn it down in 1 pass. Might work if you can set a very fine feed. Many years ago I saw the technique demonstrated at a Model Engineering exhibition – I think it was on the Myford stand with a 254 lathe. The principle is that you have the stiffness of the starting diameter, rather than the more slender interim diameters. It’s a bit buttock-clenching at first! Maybe worth testing with a very short standout from the chuck first to see whether the cut works.
An alternative to the tailstock live centre is a tailstock revolving chuck. This will need a sacrificial extra length to be cut off afterwards, but in theory it will restrict the bending of the thin section better than a centre.