That wouldn't apply if everyone included tax in their prices. In this case, the item would just be $1.10.
If the business really thinks they will lose money by pricing over a dollar, then yes, they would have to take that hit. But they are already taking that hit if the "real value" is $1.02 for example.
It's just a price/demand curve. They would simply have to optimize it differently.
This one suggests it depends on the consumer, with "highly numerate" consumers being less likely to think the price is lower (but presumably also less common in the customer pool):
Edit: searching for "charm pricing" specifically suggests it does work, but I suppose maybe there's some bias on the context in which people use that term.
If the business really thinks they will lose money by pricing over a dollar, then yes, they would have to take that hit. But they are already taking that hit if the "real value" is $1.02 for example.
It's just a price/demand curve. They would simply have to optimize it differently.