Assuming that your crooked tooth is bothering you (and not just your dentist), then there is room to permit the wearing of braces. This is easier to permit if the misalignment is affecting the function of your bite or leading to any pain.
An item that is in place for a medical indication and is left there for a prolonged time period is not a chatzitzah. Nine months fits into this definition. The fact that it can only be removed professionally also adds to its not being considered a chatzitzah.
If permitted, you need to clean it well, remove any temporary parts, and make sure it is in place and properly maintained prior to immersion. It would be best to insert after mikveh immersion so that any irritation has resolved before the next time you need to use the mikveh. You would have it inserted shortly after a mikveh immersion and would consult a rav prior to scheduling its ultimate removal.