How can a Chrome extension save many files to a user-specified directory?
Use the Native Messaging App idea.
The native app is cumbersome and a pain to write because documentation is poor, and unless you get the JSON formatting exactly correct on both ends you will not see anything in a console because stdin and stdout are taken over.
But, you will be happier when it is done because you can use standard tools (e.g., Windows Explorer, a hex editor, TeamViewer...) to view, move, and delete files, and otherwise see what is going on. Chrome's sand-boxed file system works, but seems to now be a dead-end (no other browsers have picked it up). No one is likely to develop third-party tools for it. Of course you probably don't need tools once everything is working, but until then, debugging is a nightmare because you need to write code (and quite a lot of code) just to track what files are in what directories, file versions, remaining disk space...
I am afraid that you have done your homework, meaning you looked at all possible alternatives.
The best way to achieve exactly what you want, would be (as you mentioned) using a supporting native app and communicating through Native Messaging. BTW, since bandwidth is rarely a problem on intranets, you might find it simpler to pass the resources (e.g. images) URLs and have the app download and save them.
(Yes, it will be more cumbersome than simply developing an extension, but one's got to do what they've got to do, right ?)
On the other hand, if you are willing to sacrifice a little bit of user's experience over simplicity of development, I suggest combining the HTML5 goodies (that allow you to create and download a file locally) with a JS zipping library (e.g. JSZip), so the user only has to download a single zip file (and is only prompted once). BTW, if the user wishes, he/she can choose to always download the files without prompting (but you knew that already).
You've done quite a lot of research. Indeed, regular web pages cannot write to the user's filesystem without any plugins or extensions. Also, the HTML5 Filesystem API only provides access to a virtual filesystem, as you've observed.
However, you are confusing the chrome.fileSystem API with the HTML5 FileSystem API. Unlike the HTML FileSystem API, Chrome's fileSystem (app) API can directly write to the user's filesystem (e.g. ~/Documents or %USERPROFILE%\Documents), specified by the user.
This API is only available to Chrome apps, not extensions. This is not a problem, especially since you're developing an internal tool, because you can install the app and extension, and use message passing to communicate between the extension (page action) and app (file system access) (example).
About chrome.downloads: Since your extension is internal, you can probably force users to get on the beta/dev channel in order to use this API. The only limitation of this API is that the files will be saved in (a subdirectory of) the user-defined Downloads folder.
EDIT: The chrome.downloads API is now avaiable in all channels, including the stable branch (since Chrome 31).