Dan is correct for steps i. through iii. and has successfully drawn product X (the secondary benzyllic alcohol that is the first compound on the second line of the first scheme Dan drew). Dan also successfully noted the Shapiro reaction as the name for the first transformation: conversion of a ketone to a vinyl lithium. Bamford Stevens is usually run in protic or aprotic solvents without strong alkyl lithium bases (and usually only 1 equivalent). In protic solvents, a saturated carbocation is actually formed at the former carbonyl carbon - and will react like normal carbocations do. In aprotic solvents, carbenes can be formed to do carbene chemistry.
Trapping of the vinly lithium with an electrophile is the Shapiro reaction. Dan's mechanism goes astray, however, in the acidification step. The alkene is not the most Lewis basic site on the molecule.
(and, not to nit pick, but your 'more BuLi' in your scheme is incorrect. You've already accounted for the 2 eq of base, and already have the negative charge you need to extrude nitrogen. No more BuLi is needed after explusion of the sulfonate)
I had this thought yesterday, and thought I'd try it out. I've attached the predicted NMR spectrum of the product - as predicted by NMRdb.org, my new favorite NMR prediction website. The delta1 column is chemical shift, pattern is multiplicity, and the integ. column is integration. I don't know if this will help or hinder, so let me know how it is received