I wouldn't pay for any treatment until after I got my Visa, what I would do would be to contact the orthodontist that you're dealing with, have them write out a treatment plan that explains it'll be a multi-year process that goes into detail about appointments and timings, have it stamped by them and submit that. Also likely good to submit some documentation about having some initial appointments booked.
Ask your Dr for longer, explain to him/her the circumstances of obtaining a DTV, seems like they should be well aware of your prognosis and could easily stamp something that has a longer time frame written out. Technically any embassy or consulate can easily refuse bc of 6 weeks if you're from a 60 day exempt country.