Bitcoin is completely separate from Fiat. It works without Fiat and doesn't need it.
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. How is it supposed to work without fiat, since it's historically shown that the government
has to control the money of the society in order for it to operate properly?
To answer the OP's title, I personally think that it can't work without fiat existence, because I don't believe in anarchism nor do I think that governments would regulate a censorship resistant type of money.
Even though it's often measured in fiat currencies, it doesn't mean it needs them to
work. It is entirely possible (technically) for an economy like a country or even multiple countries / whole world to use BTC as currency. Like, EUR is also measured in USD, still it would work without USD. I never calculate how many USD a banana costs when I see an € price and then decide whether to buy it.
Same could technically apply with BTC. It is possible, that people get paid in BTC and go shopping in BTC, and just "know" what a good BTC (or satoshi) price for a banana is, just like I know what a good price is in €.
Regarding government having to control money, that's a whole different topic, but I personally think it's robbery to just print money when you need it, and people who saved up money in the bank for years, and worked hard for it, have it simply devalued by a huge amount through your printing activity.