How do I force a download in the admin area?

It’s what I thought – you’re trying to send the file from the display callback, by which time it’s too late. Look for the add_*_page() call that registers the menu item in the admin sidebar – the return value of this function call is the name of the hook you can use to attach the download function:

$hook = add_submenu_page( $parent, $title, $menu_title, $cap, $menu_slug, $callback );

if ( $hook ) // Will be false if current user does not neccessary permissions
    add_action( "load-$hook", $download_callback );