Without knowing more information, I cannot say if the energy code is needed or not. If the cooler is part of the building's thermal envelope, then I would require it to comply with the energy code for insulation values. If the cooler is inside the thermal envelope, then I would say the energy code is silent in the matter as this is basically a piece of refrigeration equipment (unless the 2012 code now addresses coolers.)
As far as studs having little to no insulation values, it all depends on your point of view. Compared to the typical R-11 to R-15 fiberglass insulation fit between studs, 2x4 wood studs are inferior to resisting heat transfer. But compared to metal studs, are wood studs are superior for resisting heat transfer.
Assuming:
1) 2x4 wood studs at 16" centers
2) 15 mph wind on one side of the wall (because the original post does not indicate if this cooler is inside the building or not and there is not much difference in the R-value between 7.5 mph summer time wind and 15 mph winter time wind)
3) still air inside the cooler (assuming steady state once the fans shut off, doors are closed, and the air stops moving)
4) two layers of R 7.5 batt insulation can be fit between the studs without being compressed
5) two bottom plates & one top plate (25% wood and 75% insulation in contact with the gyp)
6) 1" air gap + other assumptions to determine an R-value for air
7) 1/2" gyp
8) no siding
9) no exterior substrate
10) no thermal resistance for SS panels
Then I calculate the overall R-value, using parallel flow and isothermal plane methods from Ashrae, to be around 12.3 and 11.5, respectively.
Switch out the steel for wood, make more assumptions about the steel thickness, flange size, and stud depth (3-1/2", 3-5/8", or 4") and the overall R-value drops to 5.7 and 2.7.