Thanks for the additional clarification.
Here's my take...when you wrote, "I worry the p-value would be deflated since we aren't capturing the measurement error...". You are onto something...but keep in mind the relationship of any observed value from a measurement system to it's two components of variance. You are indeed 'capturing' the measurement error in your data because any time you observe a measurement of something it has two sources of variation that are additive. Variance of the product + Variance of the measurement system = Variance observed. So I think what you are trying to do is come up with a way to subtract the variance of the measurement system from the variance of the product and somehow estimate a mean for each response. With only single measurements this is an impossibility since you can't estimate the mean of the product by simple subtraction of variances.
You have discovered one of the basic challenges of statistical thinking and inference in that sometimes variation of measurement systems can swamp the variation in a product/process, making statistical inferences about the product/process more problematic as the disparity between variance of the measurement system and variance of the product/process increases...in lay terms, it's harder for a signal (the product/process of interest) to rise above the noise (measurement system).
So where do you go from here? One obvious recommendation is bite the bullet and get replicate measurements from the system and use the averages of each response for modeling purposes...this will help beat down the measurement system variation wrt to estimating the mean of each response. But that sounds like it's not feasible? A next fall back position could be to build a model with the data you have...then use bootstrapping within JMP to examine parameter estimates and their distributions from a practical point of view. A last idea is within the construct of the JMP Prediction Profiler and Simulation capabilities spend some time simulating the system's behavior. There is a capability to 'add' variation to the simulation results over and above factor variation instructions so this can be a surrogate for adding measurement system variation to the mean of each response.
Hope this helps a bit?