The fix was VERY easy from what I can remember. In customer-processing-order.php, inside the tbody tags I changed the true to false to remove the sku from the e-mail.
<tbody>
<?php echo $order->email_order_items_table( $order->is_download_permitted(), true, $order->has_status( 'processing' ) ); ?>
</tbody>
Below is the code from customer-completed-order.php, I would suggest trying the same with the true / false options in it…
<tbody>
<?php echo $order->email_order_items_table( true, false, true ); ?>
</tbody>