I don´t believe the 10-22 will not work on the 6D.
I would go with Neuro´s advice and choose the 35mm f1.4 (both the Canon and Sigma produces great results). An alternative is a light travel tripod, like the Gitzo GK1580TQR5. It is extremely light and compact and it is remarkably stable considering how small and delicate it is. Downside is that it, ball head included, is about the same price as the Sigma 35.