When I publish my project the site's web.config file is correctly transformed but the settings.config, which is one folder down, isn't changed. In my published results I end up with all three files: settings.config, settings.Debug.config, and settings.Release.config.
The project seems to know that the three files are setup as a transformation since they're nested in the Solution Explorer but what's preventing the transformation from happening on just the one file?
I was able to resolve the problem with the information @Michael provided here. Specifically that only web.config transformation is supported by default and an extension like SlowCheetah is needed for other files.
My site's original creator had used SlowCheetah to set it up, which I saw in the project file, but not only did I not have the extension I needed, somehow the files weren't configured correctly either.
I installed SlowCheetah from Tools/'Extensions and Updates' but still couldn't preview the transformation. I tried to re-'Add Transformation' on my base settings.config and it triggered Visual Studio to update SlowCheetah and clean up my files. My settings.config file is now publishing correctly.