I think you can do it on DSOrganize with the TouchPod firmware (Sakura won't load DSO). I don't use DSO much, so I'm not 100% positive on that, you'll just have to try it (unless someone else reading this can confirm it for me).
Other than that, the AceKard2 allows you to cut, copy and paste files straight from the firmware GUI. There might be other flashcarts with native support for those functions, but I only know about the AK2.