We actually provide the source for the server app within Prodigy, so if worse comes to worst you could make the changes you need to the app.py -- have a look within your Prodigy installation. Unfortunately, there's no easy way to make the front-end aware of the prefix without recompiling the app.
I've definitely struggled through this before as well, so I feel the pain. What I've ended up concluding is that path-based routing is just more trouble than it's worth, and you're much better off just using subdomains. Apparently Apache has some features that allow link rewriting, so that the path-based routing will work as expected, but I usually use Nginx, and I didn't manage to get it to work.