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Interferometer "questions": clearly the mirrors can impart energy to the light and vice versa.
But I think that a part of the answer is more fundamental than this: according to Feynman the passage of a detectable gravitational wave imparts energy to any coupled massy environment through which it passes, and indeed this is necessary for gravitational waves to be detectable.
The question, then, is not whether energy is transferred, but whether it is possible without a priori knowledge of the arrival of the wave to establish a usable non-uniformity in the particular configuration of the interferometers or in the detected light. It's possible that this is the case, but it's equally possible that the light wavelength increases and reduces equally, and that (without knowledge og the arrival of the wave) the mechanical work can only be accessed as heat, and the optical effect would appear as equal and opposite variations (over time) in photon energy.
I have little doubt but that extraction in the above sense would be possible if we could both model the build-up of the gravitational wave and also detect the onset of the gravitational wave in the early part of the build-up. We are clearly a long way from either of these requirements