The shop will do the fiat conversion for you. The effect of that, the fluctuating prices for everything, remains. They can't hide that from you.
Discussion
That is irrelevant to the user seeking to live in a Bitcoin only world.
One day merchants will no longer want your fiat and will only want Bitcoin
Who cares what the shop does? If I pay dollars on Venmo to a shop in Europe and they convert to euros, or gold or even Bitcoin, what the hell do I care?