I finally found an Ant task for WebDAV that works (I used cadaver instead because in the past none had worked for me). Thanks a lot to Ole Christian Langfjæran on Github for putting this together. But I needed an additional parameter so that the password can be retrieved from a file instead of being in the Ant script in plain text. So I forked the project and did my additions.
If you are looking for a small cross-platform WebDAV client, you should probably check on DAV Explorer. It looks a bit old-fashioned but is small (less than 600 KB), does not require installation, and simply worked for me. The only requirement is a JDK and its “bin” sub-directory in the path, but you have that anyway, don’t you?
It is probably not the best choice if you are a non-technical person, want to work with it every day, and look for a nice integration into your system’s GUI. But for quick tests or the occasional development-centric work it’s absolutely sufficient.