Installing on a computer in the cloud shouldn’t be any different to installing on a computer next to you, in respect to editing playlists. But you’ll need a client that supports editing playlists, not all do.
I’m not aware of a working Amazon music backend for Mopidy.
load your playlist in the tracklist, you can then modify the tracklist (add / remove songs, reorder the tracklist), then save the tracklist back to a playlist (only m3u playlists can be saved).
Back to the other question I had, is there a technical reason that Amazon Music doesn’t have an extension? Or is it just not popular enough/no one has tried?