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“The problem is that you are assuming that the Earth's center of mass is a privileged reference frame.”

It is. A gravitational field is roughly concentric. The center is therefore privileged.

“Besides, who says the lab is on Earth? All the observer sees is an object (the astronaut) following a parabolic path with a relative acceleration of "g" with respect to the lab.”

If both the lab and the astronaut are gravitating freely, then granted, the two observations are relative.

“In other words, their circumstances are equivalent. They can each determine their relative acceleration, and if they had scales, they could determine the apparent gravity in their frame of reference. But there is neither absolute acceleration nor absolute gravity.”

If the lab is on the earth’s surface, the scale registers weight. If the astronaut is gravitating freely, her scale registers nothing. What is that if not an absolute difference?