Yeah you do, it's itemised on the receipt/invoice. You're legally required to break it down and provide a receipt so that business customers are able to correctly account for VAT. The only difference is which price goes on the tag/menu.

I've heard this argument loads of times in the US, and it's clearly based on a belief that taxes should add friction so people understand how expensive they are.

Most other countries have gone the other way and decided that transparent sticker pricing is more important than people understanding that their taxes are separate from the price of goods.