[throwaway09223]

I think you might be misunderstanding. Clocks must necessarily have a frame of reference. TAI's frame of reference is imaginary (geoid), and in reality we calculate it from an average.

Because TAI's frame of reference is the geoid it will experience relativistic effects based on earth's motion viewed from any other frame of reference (in OP's example, from the reference of the sun)

Clocks cannot exist without a frame of reference. Accurate timekeeping necessarily involves tracking spatial information as well.