To what degree does historical measurement inaccuracies contribute to climate change models?
I would estimate 0%, as long as we are considering the same types of models. See, climate models and in the regard that you may think of observations. Climate models aren't really initialized on observations of the system, but rather on inputs/forcings and mean state.
Observations of today's climate may validate the accuracy/tuneness of a climate model, but the types of observations used to run climate models may be different from what you expect. For example, the thermometer was invented either near 10 AD or in the 17th century, so it was obviously not a widespread invention then. Usually, those temperature measurements are taken as proxies, such as the amount of oxygen isotope or atmospheric composition in ice cores. So the answer to the climate skeptic is that it isn't just the temperature itself, but the modern day observations of temperature proxies that show past warming.