Looking at the Artio website, I can see their Hikashop extension only supports up to version 2.3.0, but I have 2.3.2. Maybe therein lies the problem?
You've got back-end privileges, but if it helps, I can summarise these points here:
Force a menu - yes, I've got a menu that has the hikashop checkout menu item on it, and I'm using that one here, though the menu itself is not displayed in a module.
URLs of checkout:
The first page comes up as index.php?option=com_hikashop&ctrl=checkout&Itemid=330 but also works with &task=step&step=0 as follows:
index.php?option=com_hikashop&ctrl=checkout&task=step&step=0&Itemid=330 -> using Joomsef I had attempted: checkout/cart
2nd page:
index.php?option=com_hikashop&ctrl=checkout&task=step&step=1&Itemid=330 -> using Joomsef: checkout/login
3rd page:
index.php?option=com_hikashop&ctrl=checkout&task=step&step=2&Itemid=330 -> using Joomsef: checkout/shipping
4th page:
index.php?option=com_hikashop&ctrl=checkout&task=step&step=3&Itemid=330 -> using Joomsef: checkout/payment
After payment:
index.php?option=com_hikashop&ctrl=checkout&task=step&step=4&Itemid=330 -> using Joomsef: checkout/receipt
So in JoomSEF each of these URLs were custom set to get what I wanted.
I've setup JoomSEF now slightly differently, but the problem is still there.
I've tested standard Joomla SEF and it works fine (but still looks ugly), so it appears a JoomSEF issue.