Somehow, /wp-admin/includes/template.php is not being properly connected to the wp_list_table class. I tried loading the widget after the rest of the theme had loaded, thinking it was maybe that the files were being uploading in the improper order, but that didn't change anything.
Also, the wp_list_tables class works great in my plugin's admin section. It's only when I try to instantiate the class in either a widget or a shortcode.