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Alex, I think that your suggested method is just a background subtraction which is commonly done during the image processing stage. Normally you would not take another image but use areas without nebulosity and without stars (or in between stars) in the existing image. This way the background gradients can also be removed.
The main problem with subtracting background from a different part of the sky is that the background and hence the background gradients would be different and can make things worse.
Apart from the gradients the main problem for imaging while the moon is up is that the weaker signals will get swamped by the increased background and cannot be distinguished from the background noise any more. That's why the dark skies work the best.
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