To me this seems like a good example of when not to use a Web based solution.

I have a small personal project — a Web application that should be run locally by the user. That probably makes it a desktop application,

It probably does yes.

but the interface is through a browser (it kinda makes sense to represent this application as an interactive document). So I run a web server on a port with a large number.

If it makes sense to represent the user interface as an interactive document it does not necessarily mean: web based solution. As far as I know most GUI’s around have widgets for representing documents as well. Take for example products like oXygen and XMLSpy: Fat clients in which the interface can be presented as an interactive document. I am sure there are many more examples.

My advice would be to rethink your solution. It would save you a lot trouble.