Forgive me if I'm stupid and I missed some obvious documentation somewhere, but can anyone tell me how I can set up my own non-gui-related Perl/Tk event to handle this?

You've got the magic word in your title, and it's "callback", looking at the Tk documentation you should find Tk::callback, which in it's "SEE ALSO" section has "Tk::bind Tk::after Tk::options Tk::fileevent". Tk::after is what you're after ;)

Also useful is super search and http://perltk.org