I s'pose the alternative would be some sort of direct measurement of the road surface; eg, ultrasonic, radar, or optical - maybe doppler?

GPS?

I doubt that there'd be a simple relationship to pressure: an under-inflated tyre doesn't just have a reduced circumference - the whole way it deforms under the load during rotation changes.
The case of over-inflation is probably simpler?

The effect probably also depends upon the vehicle loading?

And don't forget that tyres warm-up in use - especially when under-inflated - which would tend to increase the pressure.

In conclusion, I'm with Erik: if you really need a precise measurement, you can't rely upon the vehicle's running wheels - you need something independent, which you can specifically control to the required tolerance.