Most half-pound foam installers going for a stud fill will blow some excess and trim it flush. Not so in this case, eh?
Sheet goods on the exterior are a good option (and rightfully shoulda been in the retrofit plan in the first place). By putting the foam on the exterior of the sheathing it more than doubles the R-value at the studs & framing, which would otherwise have been a thermal short-circuit. Assuming the wall is 2x4, 16" o.c. construction, the center-cavity R-value of the Icynene is ~R13, but at the studs it's only ~R4 (max). Add ~R1 for the sheathing and it's ~R5 at the studs, R14 center cavity. But if you add 1" of XPS (R5) to the exterior it adds R5 to the studs (cutting the heat loss there in half) and brings the center-cavity to R19-ish.
The "whole wall" and "clear wall" R-values go up considerably when you get rid of those sub-R5 short-circuits from framing elements. Even if some of the Icynene is only half-depth, and inch of XPS will more than make up for it. I'd insist on R5 as a minimum, since there are voids and gaps, but you'll end up with decent perfromance if you do- at least as good as if they'd done the job perfectly in the first place. R2.5-3 (half-inch XPS) might work too, depending on just how much of it is screwed up. Push for R5-R7.5, but don't settle for anything less than R4 (1" of EPS).
Play around with
this calculator developed by the Oak Ridge National Labs, which should give you a good feel for what the exterior foam will do for you.
Select either cellulose or R13 fiberglass for the cavity insulation type- the polyurethane foam in their calculator is the higher density closed cell stuff, not Icynene. With "no foam" selectected in the calculator this would be more or less where you would have been if the job had been done right initially. Select R11 for a best-case of how it probably is as-patched-with-voids, etc. If they don't want to spring for more than R3 (which may be reasonable, depending...) it's still cost-effective in MA to offer to pay a bit more for the material costs to bump it to R5. The labor costs will be the same for either, but the material will roughly double. (Make them show you their material/labor breakdown if you go this route.)
Stick with XPS if you can, but if they want to use EPS or ISO, insist on un-faced or fiber-faced breathable stuff, or you may end up with condensation issues within the cavity. Icycene at 3.5" is pretty vapor-permeable, and best-practice in most of MA would be to use at least vapor-retardent paint on the interior finish wall surface and let the assembly dry toward the exterior. If they use foil or poly-facers on the rigid board the sheathing layer can't dry in winter, since it's inside the vapor -envelope, and colder than the dew point of the interior air much/most of the winter. XPS is semi-permeable, which allows the assembly pass water as vapor out to the drier winter outside air. Fiber faced iso or un-faced EPS is even more vapor permeable than XPS, but not as structurally rigid, easier to damage in handling, etc.