I find it very, very hard to believe that you could see such big difference. PV=nRT and all that. Have you really measured that big of a difference? If you are seeing that with the Tirewiz, I'd seriously question it's accuracy.
climbing 5000'
from sea level would be a drop of approximately 150mb. So I can see a 2-3 psi pressure difference in a cold tire that was driven up to altitude, as measured by TyreWhiz. Note that at altitude climbing 5000' creates
less pressure difference than at sea level.
If the sensor is truly measuring an 11-12 psi pressure difference, that tire got seriously hot - or the cyclist climbed into a hurricane.
How hot? The equation you wrote can be re-written as P/T= nR/V where nR/V is held constant, at which point it becomes Gay-Lussac's law. Taking the inflation temperature to be 20C, we get (23+14.7)/293 = (35+12.7)/x
or 47.7/x = 37.7/293
or x= 293*47.7/37.7
It is far easier for me to believe that TyreWhiz is off than that the tire is water-boiling hot.