[Babcock_Hall]

I have found a number of syntheses of vinylsulfones using Horner-Wadsworth-Emmons (HWE) chemistry.  Some of them start from diethyl ((phenylsulfonyl)methyl)phosphonate.  I have only found a few examples in which the α-carbon bears an alkyl group, and all were in Ellingsen et al.* "Sulfones in Horner-Wittig-Synthesis of Alkenes" 1979 Acta Chemica Scandanavica B33:528-530.  In this paper diethyl ((phenylsulfonyl)methyl)phosphonate was first alkylated to put a methyl, ethyl, or benzyl group on the α-carbon using sodium hydride as the base.  The product of this reaction was then treated with an aldehyde.  Methyl iodide or primary alkyl halides were used, but my reading is that the alkylation did not work when the alkyl halide was secondary.

My questions are (1) whether or not one could alkylate using ethyl bromoacetate and (2) whether or not one would expect problems in a subsequent HWE coupling.  With respect to the first question, I tried SciFinder, but I may or may not have searched in an optimal manner.  I am happy to provide reaction schemes or further references if either of those would help.
EDT
*A paper (Todd Blumenkopf "An efficient preparation of diethyl phenylsulfonylmethylphosphonate" 1986 Synthetic Communications 16(2):139-147) described an analogous reagent bearing an allyl group at the α-carbon (4c), and the paper seemed to be a prelude to HWE chemistry.  The compound in the title is 4a in the paper.

[rolnor]

I think you just have to try it, it could work. The sulphonylphosphonate should be a soft good nucleophile. One could worry about the alpha-proton in ethylbromo acetate being acidic but its hard to see for sure if it will pose a problem I think.

[Babcock_Hall]

Clayton H. Heathcock, Todd A. Blumenkopf, and Karl M. Smith, "Total Synthesis of (±)-Fawcettimine" J. Org. Chem. 1989, 54, 1548-1562.  https://pubs.acs.org/doi/pdf/10.1021/jo00268a015

In the JOC paper compound 28 is the α-allyl sulfonylphosphonate 4c from the Synthetic Communications paper in 1986.  The authors of the JOC paper attempted to react it with aldehyde 17 using butyl lithium as the base without success.  This aldehyde (17) has several functional groups, which I suppose might have interfered in some way.  Yet the authors also tried to react 28 with benzaldehyde and did not obtain a product that they could characterize (footnote 23).  Heathcock and his collaborators cited Ellingsen's paper, in which the HWE chemistry was successful.  Perhaps the allyl group caused difficulties not encountered with methyl, ethyl, or benzyl groups.