Hello,
I am now getting an "add to wishlist" button instead of an add to cart where the product is a variant
You should have an "add to cart" button, so can you check that your variant is in stock ?
Also, the parent images etc are not available for variants
Yes, that's the problem with the solution that Xavier gave you, it will require more than this to display every information related to your variant.
To display your variant name the solution will be to directly write it on your Joomla article
To display your variant image the solution will be to directly add an image through your variant configuration page.