Cost of living isn't the only factor. Lifestyle quality is another, and not fully captured in the model you give because unlike money, people are non-homogenous. This may be more evident when the demand for a given field in a region is particularly small.
Consider [0] says that petroleum engineers are one of the highest paid STEM jobs, and [1] concurs. But a petroleum engineer in, say, North Dakota where there's an oil boom is likely going to have a higher salary than one in Hawaii even after adjusting for local wages. There are so few petroleum engineering jobs in Hawaii because it has no oil resources. If 0.1% of oil engineers are are willing to take a pay cut to move to Hawaii, then that small subpopulation may be enough to supply the Hawaii jobs market. (1 out of every 1000 jobs in ND is a petroleum engineer. We know that wasn't supplied by the indigenous population. While BLS doesn't have an entry for petroleum engineers in Hawaii, indicating that it's small.)
In a related vein, [1] points out the nuclear engineering is the highest paying job in Tennessee. This is probably influenced by Ridge National Labs and the three reactors in the state. The University of Tennessee offers a MS in nuclear engineering, so there's definitely a local production of people with those skills.
While Montana (no reactors, no nuclear research) doesn't have a BLS listing for nuclear engineers, which indicate that the demand for such is quite low. Low enough that secondary influences, like the desire to be near good fly fishing, or to move back to family in Montana, may be a bigger factor than cost-of-living.
Thus, regional salary variations, even when adjusted for CoL, may not give the estimate you're looking for.