I have the prices set to be inclusive of tax but if not logged in, then the tax amount will be added to the total in the cart. If you are logged in or a cookie is set to remember your location, then you will see the price of the product plus tax throughout the site. If I am correct, then the price shown through the site and in the cart shouldn't change besides when shipping is added.
I have another WooCommerce site that has taxes setup this way and has no issues, the only difference I can see is I'm not using WPML on the other site.
I have already switched to the base Storefront theme and disabled all plugins but the issue still remains.
I will provide access, but please use the Ireland and Netherland versions of the site to debug as the others aren't setup for sales.