You're doing nothing wrong, but Tiled is intensionally not allowing you to edit the properties of external tilesets, and this is indeed because of a bad design (such edits would currently never be saved). To work around this, you need to temporarily import the tileset, edit the properties and then export it again. In this process you overwrite the .tsx file, thereby saving your changes.
Eventually, it would of course be nice to fix the design so that edits to .tsx files are properly tracked, so that they can be undone and saved.
Regards,
Bjørn