First, welcome to the community. Interesting question. Here are my thoughts.
My first thought is how much measurement error is there? It may depend on how much measurement error there is.
Can you use averages to reduce the error? I suppose you could use different values of the extremes of the measurement error distribution as different Y values (e.g., lowest, largest and midpoint) and model each to see how that impacts the significance of model effects.
Usually you design an experiment with a model in mind and then as a result of the analysis, you reduce the model by removing insignificant terms and then iterating.
Perhaps others will have some ideas for you.
"All models are wrong, some are useful" G.E.P. Box