Is there a way to add another row to the tinyMCE kitchen sink toggle?

Yes!

• Use the mce_buttons_2 filter to add
  buttons to the second row.
• Use the mce_buttons_3 filter to add buttons
  to the third row.

Here’s an example of what I use:

function mytheme_mce_buttons_row_3($buttons) {

     $buttons[] = 'fontselect';
     $buttons[] = 'fontsizeselect';
     $buttons[] = 'code';
     $buttons[] = 'sup';
     $buttons[] = 'sub';
     $buttons[] = 'backcolor';
     $buttons[] = 'separator';
     $buttons[] = 'hr';
     $buttons[] = 'wp_page';

     return $buttons;

}
add_filter("mce_buttons_3", "mytheme_mce_buttons_row_3");

Just drop this in functions.php. (I put it in my Theme setup function, that gets hooked into after_setup_theme.)

EDIT:

I don’t know if it makes a difference or not, but you’re using array_push($buttons, $button), while I’m using $buttons[] = $button

Here’s your code:

//setup array of shortcode buttons to add
function register_button_3($buttons) {
   array_push($buttons, "dropcap");
   array_push($buttons, "divider");
   array_push($buttons, "quote");
   array_push($buttons, "pullquoteleft");
   array_push($buttons, "pullquoteright");
   array_push($buttons, "boxdark");
   array_push($buttons, "boxlight");
   array_push($buttons, "togglesimple");
   array_push($buttons, "togglebox");
   array_push($buttons, "tabs");
   array_push($buttons, "signoff"); 
   array_push($buttons, "columns");
   array_push($buttons, "smallbuttons");
   array_push($buttons, "largebuttons");
   array_push($buttons, "lists");     
   return $buttons;
}
add_filter('mce_buttons_3', 'register_button_3');

Which, using my method, would look like this:

//setup array of shortcode buttons to add
function register_button_3($buttons) {
   $buttons[] = 'dropcap';
   $buttons[] = 'divider';
   $buttons[] = 'quote';
   $buttons[] = 'pullquoteleft';
   $buttons[] = 'pullquoteright';
   $buttons[] = 'boxdark';
   $buttons[] = 'boxlight';
   $buttons[] = 'togglesimple';
   $buttons[] = 'togglebox';
   $buttons[] = 'tabs';
   $buttons[] = 'signoff'; 
   $buttons[] = 'columns';
   $buttons[] = 'smallbuttons';
   $buttons[] = 'largebuttons';
   $buttons[] = 'lists';     
   return $buttons;
}
add_filter('mce_buttons_3', 'register_button_3');

Give that a try?