I've figured it out. You have to go to
cart slash admin slash controller slash response slash sale slash invoice.php
Inside:
$this->data['orders'][] = array(
Look for the line:
'total' => $total_data
After the last line in orders add a comma and pament_method_key so it looks like this:
'total' => $total_data,
'payment_method' => $order_info['payment_method_key']
That will add the appropriate data to the orders variable before it gets passed to the invoice.
After that go to edit
cart slash admin slash view slash default slash template slash responses slash sale slash order_invoice.tpl
and add the line:
<?php echo $order['payment_method']; ?>
Where you want the payment method.
Apologies for the awkward formatting; this is a new account so I can't post links or code.