Two things come to mind you can check:
1) Does your Checkout setup in HikaShop Configuration > Checkout include the ,end step? Checkout breaks if that is missing
2) Is your payment method restricted in any way to a user group or zone? Check the configuration of the plugin.
If the store is offline, then to test you are presumably logging in as an admin on the frontend. As you've seen, potential problems can be hidden by testing this way under an admin account. It's good that you are doing final testing before go live with the site online on its dev URL, so everything will work as it will appear to the public after go live. Beyond that, we make it a policy to never use admin accounts for testing, which means you need to toggle the site online briefly during testing and then put it back offline. It's a bit of a pain to have to toggle this back and forth, but superusers are all powerful. Using a superuser as a development testing account can give you an inaccurate view of what is and is not really working.