This opens up a big discussion. If it's separate from fiat, then why do we measure bitcoins in dollars, euros, yen and not the opposite? We could do both, but we would never measure a dollar in sats.
We do this probably because fiat currencies are already the staple means of exchange and every other thing is valued against it. Cars, gold, houses, stocks are all priced against fiat. This does not mean any of this assets cannot be valued against each other, stocks can be sold for gold and houses for Bitcoin, they are not just popular enough or scalable enough to be used daily for all transactions as yet. In the future Bitcoin could be valued against something else, or other assets valued against it.
To answer the OP's title, I personally think that it can't work without fiat existence,
Not yet.