Thanks for sharing - this is interesting. Did you turn off the option on the Linux side to install automatically missing modules (assuming that such an option exists there, too)? One of the nice things about MikTeX in Windows is that the modules are loaded dynamically inside of LaTeX (or even LyX). TeXLive has huge "chunks" for modules (via apt-get), at least as far as I can tell.
If you turn it on, then missing packages might be installed twice. I am not sure if you can configure MikTeX such that packages are installed in the common package directory (if I remember correctly, you cannot move it up above the Linux root directories in the MixTeX console). If you turn it off, then you must either install the packages by hand or switch to Windows, which may not be such a problem when using WSL. You might also try to add the Linux package directory to the list of root directories under Windows; I haven’t tried this but perhaps it works.
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