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Edit: and if you are dealing with real data sets or producing real datasets for analysis you will often have only approximations to the thing you want to measure. Determining whether your proxy variable is worth including or how to interpret your results in light of it are necessary skills to develop.
I get that invertibility means that you can't fully recover the original racial percentage, e.g., that a 48/52 split and a 78/22 split will both look exactly the same, since (.48-.63)^2 and (.78-.63)^2 are equal. I don't see why that totally taints the entire dataset.
If he doesn't understand it, then it's not in bad faith. Right?