As a Chemical Engineer, I thought at first glance that your calcs must be off. So I dusted off the ideal gas law equations and ran them for myself. I confirmed your temperatures for a given pressure rise.
I see pressure rises on my TPMS but not quite that extreme. On the other hand I haven't driven across the desert.
Thinking about this some more, and knowing I never see tire temperatures over about 140 using an IR gun, it hit me that one thing we haven't accounted for is the ambient air rushing over the tire and cooling it as we run down the road.
Conclusions, one the air temperature may indeed be that high, but the rubber is cooled by the ambient air. Two, You did the right thing by adding some air to prevent a problem on the road.
