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Hi Chris,
I think that you could find debate about that topic, for sure, and probably with more learned people than I. :) All I know is that I've been told that acceleration such as that would not directly cause a time dilation effect. I've been told that the time dilation effect would result only due to the velocity gained during/by the acceleration itself.
If it were otherwise, and acceleration had an effect like such, then wouldn't the Moon undergo gravitational time dilation, kinematic time dilation, and this "acceleration" time dilation?
I ask this because (and I think you agree) that gravitational time dilation isn't really from the acceleration per se, but just from being in the gravitational field itself -- like, a test particle at the centre of the Earth would not undergo acceleration (Newton's shell theorem), but it would definitely undergo gravitational time dilation because the Newtonian potential at the centre of the Earth is non-zero.
So basically, you're proposing three kinds of time dilation? I imagine that the effect would be smaller than the kinematic time dilation, because you'd probably also have to consider a time dilations from the 1st derivative of acceleration, 2nd derivative, etc. It's an interesting thing to think about anyway, even if most people have told me "no, not possible".