Hospital stays are a whole different thing from more routine procedures. A lot of the costs are directly billed by the doctor and bypass the hospital, and hospitals don’t know what their own costs will be.
If you have insurance that covers this, it’s simple. You don’t even need to call anyone. You’ll pay your deductible.
If you want to compete on price, you’ll either need to charge substantially less than people’s dedictibles in order to make a noticeable difference, aim your price competition at insurance companies (who pay vastly lower prices already), or target the not-exactly-lucrative uninsured population.