XDP was a fairly basic mod of Dream Passport (the Japanese Dreamcast web browser) that basically just replaced text and graphics with stuff in English so that non Japanese speakers could actually use it. There's really not any major modifications (that I'm aware of) to it that would be anywhere near on the level of what would be required to make the browser able to save stuff to an ATA device or an SD Card.
That would require code modifications that are likely impossible to make (since Dream Passport was obviously not open source). It'd be easier to port some small open-source web browser over to the Dreamcast using KallistiOS than it would be to actually retrofit that kind of functionality onto Dream Passport.
That said, the real question is why someone would want to do that. It's much easier for just about everyone to just use a browser on a PC to download files, so there's very little purpose to doing so on the Dreamcast. The same goes for downloading emulators or homebrew games to an SD card -- why not just do so on your PC? If that's really all your after by doing something like this, it'd be much easier to spin up some sort of thing like one of the various "Homebrew Browser" tools that have been available on other consoles (like these ones for the Wii
, for instance) than porting over a whole web browser.