We typically advise applicants to apply in the first couple rounds, because that is when the majority of the class is decided. If you feel, however that you can put together a great application for the third round, then you should go for it. It sounds like you would be a competitive applicant.
Here's what Stanford specifically says regarding your question:
Is there an advantage to applying in Round 1, 2, or 3? Does Stanford have a pre-established quota or number of people who are accepted in each round?
The simple and honest answer is that you should apply when you are fully confident that you can submit an application that accurately represents your talents and strengths. You are investing a lot of time and money applying to business schools, so you do not want to do anything to disadvantage your candidacy, such as submitting a weak application just to apply in an earlier round. To become the strongest candidate possible, take the time to put together a thoughtful and thorough application, even if it means applying in the final round.
While we admit outstanding individuals in all three rounds, there are some advantages to applying in either the first or second round: (1) early notification; (2) the opportunity to meet other admitted candidates during Admit Weekend; (3) the opportunity to register early for on-campus housing, especially at the Schwab Residential Center, which is in high demand; and (4) ample time to plan for your transition to the Stanford GSB. There is no advantage with regard to receiving fellowships or loans from the Financial Aid Office.
At the same time, you know best when you are ready to apply. While it is true that fewer people are admitted in the final round than in the first two, we do admit excellent candidates in the third round (including our current Director of MBA Admissions). If you are not admitted this year, you are welcome to reapply the next year.