I’m assuming the theme calls some standard function to get a list of
content types, and that I need to do something to get my type on that
list.
It’s impossible to say without seeing the original code for creating options, but if I had to guess I’d say it’s likely that they’re manually entered. Any use of a standard function to list post types would probably already list yours.
So while it might not be possible to add your post type as an option, you can use the theme_mod_{$name} hook to programatically add your post type to the value of the theme mod so that the logic in comments.php is true for your post type.
The way this filter works is that you add a filter to theme_mod_comments_display and use the callback function to modify the value. Since the value appears to be an array of post types for which to show comments, we want to add episode (or whatever the actual slug is) to the array.
function wpse_299549_theme_mod( $value ) {
// Make sure $value is an array.
$value = (array) $value;
// Add custom post type to array.
$value[] = 'episode';
// Return modified value.
return $value;
}
add_filter( 'theme_mod_comments_display', 'wpse_299549_theme_mod' );