The lighting looks fine, so I don't think addition flash would be helpful. I would use a minimum of F2.8 @ 1/200th, adjusting the ISO to obtain a good exposure. With a Canon 6D, a well exposed 1200 ISO should not be a problem. I would agree that you should set the camera on M, and not be locking exposure. For Canon, activating focus using the back button as you described is fine, but I would shoot the image immediately once focused (to minimize movement of either camera or subject). At f2.8 with that lens, any slight movement by you or you subject can result in an OOF image, especially if you evaluate the image on screen at 100%. Also note that the AF point may be larger than what is indicated in your viewfinder, making more difficult to get precise focus on the eye. As suggested, using a tripod would be helpful, and would also allow an effective use of use of live view to obtain critical focus. It would also be helpful to test AF vs live view focus (on tripod) to check that the AF and lens are well calibrated.
Shoot a bunch of images and choose the sharpest/best composed ones as your keepers. Don't sweet a few OOF images that will occur when shooting at wide apertures.