If, as your model reflects, that the canopy is only some form of cloth and the 6x6 is only acting as a connector, this should be able to span 12' without sagging over time. I would suggest a timber free of heart if possible and "bark side up" (camber up) orientation.
I would also suggest reviewing some of the more "Asian" designs of structures as these do not use "oblique bracing" and have the more modern "rectilinear" format of modern design. A post, connecting beam, and 貫
(Nuki-bracing beam) configuration may be more to your liking and solve some other inherent issues within your design.