I've thought about this a bit though I haven't implemented it yet.
First, you don't use Leadwerks's publish system since Leadwerks doesn't support this yet. You need to organize/zip files yourself to guarantee you have the system you need.
Next, I would have the latest files and maybe a file list uploaded to a website.
Finally, I would use a library like curl to compare local files to online files and download and update them accordingly.