Does shortening the path length of an excited etalon do work? What about LIGO?

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

In LIGO and equivalents of course the mirrors or detectors can be considered to be moving and thus cause the change in the interference patterns, depicted as the waves we've seen graphical depictions (and sound) of. This movement is kinetic energy on mirrors or detectors, and so conceptually we get the electrical signals. These have energy, which came from the gravitational wave. The energy derived is the signal power times the detection time for each detection. Divide by noise energy and you have SNR. so, unless you have that power or energy transferred to some of the mass of the interferometer you have no detection. You can think of these waves as you'd think of electromagnetic waves, at this level of approximation, basically linearized gravity.

If you want to extract anything useful you'll have to create a huge collection area for the gravitational wave. Otherwise you can extract but it is a very small amount. Figure out the energy density, eg for the 2015 merger event, 3 solar masses 1.3 billion light years away. Energy density = 3/(4 pi X 1.3^2) in solar masses per square light year transform to joules per square meter, and you'll see how small. You'll have to increase the area of the absorbing masses to get to a reasonable amount of energy, and make sure you have an efficient mechanism.

The effective mechanism could conceivably be the same as the interferometers used, except much larger such that a strain of 10^(-11) or so, as observed in that case, could give you relative macroscopic motion you can turn into enough joules, or continuous watts, by coupling the detectors/mirrors to some hearing or electric or other conversion system. you'll want to make sure the wave arrival angle is perpendicular to your baselines to maximize the power. You may be able to do other things.

For etalons, since the movement is the conceptUally relative displacements of the etalons walls, you'd have to figure out how to extract the energy from that movement (assuming they can move differently, the stress will be real, not imaginary, so you may need to do virtual etalons or elastic or something). Maybe you find a piezoelectric material that converts that into electricity, I am not sure. But the energy is there.

Instead of interferometers you could have a circularly arranged set of bodies, separated by large distances, with the circle being being flattened and expanded in perpendicular directions consecutively at the wave's frequencies, and somehow figure out how to extract the energy from those oscillating bodies, again the heating or electric or other mechanism to convert that energy.

Either way, and you may figure out other configurations, you'd have to, seems to me, to be doing this over astronomical or astrophysical distances to get anything reasonable. Figuring out how to convert the kinetic energy to heat or electrical energy or something else you will have to think through.