Hmmm... if all things need to access the Curses::UI object, you might consider creating giving each of their objects a reference to the Curses::UI object. It should be the same object (so you don't have multiple ones), but then every time you call a method, you already have it in the object data. In that case you don't need to pass it around.

Sometimes the Singleton design pattern can solve this little sanfu, too. There is a Perl Singletons on Perl Monks, or my article about Singletons in The Perl Review, Issue 0.1