The dominant company might not have agreed to decent terms. After all, they own the market so they can starve a startup out. Though you are right, you'd think they would want to sell an improved syringe at huge margins.
Go to the nurse unions. Perhaps the next contract could require the syringe.
Go to congress. Lobby. While you may be outspent, that isn't everything. You can also apply pressure via an ad campaign, asking voters to demand that nurses are safe. There is some real emotional appeal potential here.
End result: the dominant company is forced to cave in some way. They can buy the syringe, allow an exception for the syringe, or just lose 100% of their customers.
