I *think* it might be possible if you host the private wiki instance on a node.js version of tiddlywiki and wrote a few custom node scripts (and deployment scripts) to generate the public version and then push it out to the public server.
I do something similar without all the scripting but the public wiki is it's own instance of TiddlyWiki in a sub directory of the private one. More specifically, my publicly accessible server (a hosted VM) runs a web server with 2 Apache Virtual servers.
The Apache virtual server for the private access runs on HTTPS and requires authentication. It runs the Apache WebDAV module. There is a sub-folder on it that could be called "public".
The second Apache virtual server is the public access server. It's document root is the public folder of the private instance. I use separate sub-domains, one for each virtual server.