A competing product (Clio, see clio.com) has a web-based UI and has solved this problem. When you click on a document stored by Clio, a native application program launches on your computer to open the document. When you close the document in the application program, the revised document is stored back in Clio. I don't know how they manage this, but I tested the product when I was looking for docketing software. Unfortunately, their docketing software is not designed for patent practitioners, so it wasn't suitable.
Clio's web site describes two facilities: Clio Drive (apparently a virtual drive - "Create, access, and share your matter-related files right from your desktop and securely sync them to Clio using our virtual drive experience. No need to download or reupload. Open files from the web for easy document editing.") and Clio Launcher ("... to easily open files from your web app into programs like Microsoft Word, Adobe Reader, and more. Any updates you make will be saved on your desktop and in the cloud.")
Another competing product (maybe DocketTrack or maybe another package I tested, but I can't remember which one) also solved the problem. This package requires running a (I'll call it a "helper") program that seems to communicate with the server, so that when you click on a document, you select "download" or something like that, and the server sends the document to the helper program, which then launches a local native application program to edit the document. Once the application program closes the document, the helper program sends the modified document back to the server for storage in the cloud. I found AppColl superior to the competing product in many ways, except for this ability to handle documents.
I have suggested (including screenshots) to AppColl that they investigate operation of these competing products and consider building similar functionality.
-George