@layman,
No.
(And now that I have answered your question, I will try to explain to you why you are misunderstanding the physics).
Both clocks start out in the Earth frame of reference. This is an non-inertial frame and is rotating about it's axis. From any inertial frame, the Clock attached to the Earth will show an acceleration toward the center of the earth and an acceleration toward the sun.
The airplane going east is traveling along with the clock on the earth, it's speed an acceleration wrt the earth clock is less than the airplane going West. (notice I am using the "wrt" term you used in an attempt to help you understand).
This experiment ends up being measured in the Earth's frame of reference (which is as valid as any). The easiest way to understand this experiment is to do all the calculations wrt the Earth.
You can do the calculations from other frames of reference. If you had a clock that was fixed to be motionless in the CBR frame, you would get a different answer.
How do you consider this problem Layman? Do you consider the Earth to be motionless? Or do you think that we should consider this as if the clock were fixed to the CBR frame?