That's what happened when the Euro took over all the other currencies in Europe. Stores had two prices per item, one in Euro and one in their sovereign currency. People, over time, became accustomed to the new unit of account.
Some places may have discounts for individuals who prefer to pay in Bitcoin or an additional charge for those who choose to pay in fiat.
Bitcoin is a parallel system, so it is a natural evolution of our current value system.
Personally, after a few years of using Bitcoin as my personal unit of account, I recently started to think of prices in terms of sats. My brain hasn't fully adjusted out of denominating value in dollars, but it's getting close.
It reminds me of when I learned English. I would think and count in French, then translate to English. Now, I think and count in English exclusively.