I'd like to be able to just find another screw closure just like the one on the necklace, screw it in, and then I would be fine, but I haven't been able to find one. This is a Gucci necklace, I would prefer not to have to cut off the original closure and add in ball chain. Any ideas?
From the photos, it looks like there is a spring ring clasp attached to the necklace. You could attach the extender to that instead, since it would just need to have a basic ring ending (easy to find). It might look funny with the screw ends hanging there, though.
Another idea, if you can't find the same size barrel closure, are you ok with (possibly) permanently modifying the necklace? You could either glue or solder another clasp and/or the extender, using the screw ends as a base. Solder would be sturdiest but probably permanent. Depending on the glue you might be able to remove it later, but you still risk damaging the finish with the chemicals (particularly acetone).