The procedure that you likely need is a septorhinoplasty, which is surgically fixing the internal and external aspects of the nose. The parts of the surgery that improve your breathing will likely be covered by insurance, and your insurance company will likely make that known prior to any...
By recovery, that depends on your endpoint: look not bruised or swollen, be able to swim and play basket ball. Certainly a year for the latter and 4-6 months for the former, for most people.
I would say this is a tall order for the surgeon as the bump will appear larger when the nose is reduced. I would not recommend leaving a bump. It would look like a mistake.