I think you won't have any luck by getting another VAIO Duo 11. I got the same problem with screen flickering. It is obvious that many owners are describing the same issues and no one could get rid of them.
I read many articles in the web dealing with backlight flickering of LED-driven (IPS-)screens, some examples of devices suffering from backlight flickering are the MacBook Pro of 2007 and the Lenovo X220.
The reason for flickering backlights of LED-screens is the so-called "pulse-width modulation (PWM)": When not at full brightness, the backlight is naturally changing between the state "on" and "off" very quickly. It is "dimmed" by rising the amount of time between the two states (for the human eye it seems that the brightness gets lower), so that the flickering gets more and more visible by lowering the brightness towards zero.
Whether the flickering is noticeable or not depends on the user's eyes and how the PWM-frequency is set by the manufacturer.
Some manufacturers are driving the backlight-PWM at high frequencies so that even very sensitive persons can't see the flickering. I think, SONY chose (by chance?) a very low frequency.
In the past, Apple could change the PWM-frequency of the 2007 MacBook Pro by releasing a firmware/BIOS-update, so I hope that SONY gets aware of this problem an will release an UEFI-update as well.
Besides that, there is only one solution: Set your brightness at 100%, so the backlight is on all the time. I experienced as well that changing to 40 Hz can remove the flickering behavior, but I don't know why. Perhaps, PWM-frequency and refresh rate don't interfere then.
Sorry that my (technical) english is not quite good, I hope you understand what I'm writing.