|Some War of 1812 Surgery Tools|
This possible explanation does not seem to be accurate. For starters, patients often passed out during major surgery, such as amputations. In addition, surgeons would be unlikely to give patients a bullet to bite on since they could easily swallow it. Clearly, choking on a bullet is not conducive to healing a patient.
Another suggested origin for the phrase comes from the Indian Rebellion of 1857. The story goes that a group of soldiers recruited by the British, the Sepoys, refused to fight when a new rifle design was issued to them. The new rifle used a greased paper cartridge that the soldiers would need to bite in order to use. Many soldiers refused to do so because the Hindu soldiers feared the grease was made of cow fat and the Muslim soldiers feared that the grease was pig fat. The theory is that soldiers were told to ignore their religious beliefs and bite the bullet.
After looking at a number of different possible answers for the origin of this phrase I can't definitively say where the term came from but these are a few possibilities. Either way it is a good thing we don't use it today in the above circumstances. If you are procrastinating on doing your work and are encouraged to bite the bullet, just be thankful that you don't actually have to bite on one.