How to use an omega template with hook_theme
I solved this using this question as a basis to work from https://stackoverflow.com/questions/4100485/register-your-theme-functions-in-drupal.

The template (.tpl.php file) goes in the omega template directory. You then use a hook_theme function to define the theme with the template key in the array pointing to the name of your template file.

The theme is then called with theme(your-theme-name)

Here's my working example for my module called vis:

function vis_menu() {

  $items['images'] = array(
    'title' =&gt; 'Images',
    'page callback' =&gt; 'image_page',
    'access callback' =&gt; TRUE,
  );
  return $items;
}

function image_page() {
  return theme('images');
}

function vis_theme() {
  return array(
    'images' =&gt; array(
      'variables' =&gt; array(),
      'template' =&gt; 'vis',
    ),
  );
}