According to Wikipedia, the wavelength of a gravitational wave is about 600,000 km. The Earth moon distance is about 384,400 km. Considering that the latter is roughly half the former, it seems to me that there would be an instant during which the space between Earth and moon were almost entirely squeezed or stretched as a gravitational wave passes through the Earth moon system. I'm curious why the lunar reflectors cannot be used to detect those waves more cheaply and easily than LIGO, the kilometer long dedicated interferometer that actually succeeded in detecting them. To do so, it needed to detect space-time variations on the order of a hundredth of the diameter of the nucleus of a hydrogen atom (2.40×10−15 m). Considering the ratio of 600,000:1, wouldn't a retroreflector detection require about 600,000 times less resolution/accuracy than LIGO (1.44x10-9 m), achievable without a super-sensitive interferometer?
There are a few issues with what you've described. In no particular order:
The length of the arms determines the frequency of wave that can be detected. So, for example, LIGO (ground-based) and LISA (proposed space-based) have vastly different length of arms. That doesn't change the resolution, it changes which frequencies each detects / would detect. This is analogous to changing the length of your antenna for electromagnetic waves. You need a different size antennas for different frequencies of radiation. That does not by itself say how good your amplifier has to be to detect small signals in any particular frequency.
Related, you'd have to check to see if there are plausible astrophysical sources of gravitational waves at the frequencies to which your moon-reflector system would be sensitive. (I don't know the answer off-hand because it's not a length that I've thought about much.)
You still need extreme resolution. I'm not exactly clear what you meant with the ratio you stated, but you seemed to think you could have a less sensitive instrument because of different length arms. That's not the case - you still need very sensitive measurements of change in distance. Ala what's above, you've changed the frequency to which you're sensitive but not necessarily the tolerances on the arm-length change measurement.
The moon is seismically active. I'm not expert on moonquakes, but I think it's likely that it's active enough to blow out your tolerance on the measurement, keeping in mind the extreme measures that LIGO takes to isolate itself from seismic noise and the fact that moon reflectors were put there in the 1960s with no thought for this type of precision.
You need a differential measurement in different directions. LIGO uses two arms. LISA has three legs, although interferometry is done pairwise on them. You'd need to put something in orbit at a comparable length in order to have your second leg. That's its own engineering feat. (Not necessarily impossible, but not as ready to go as you might hope.)
Assuming you are sending the laser beams from earth, you need simultaneous line of sight from the ground station to both the moon and the end of your new leg. That's going to be a problem if you want continuous coverage.
You'd need an extremely good model of atmospheric effects on the propagation of the laser beam as these are probably bigger than the change in distance that you're trying to measure. (LIGO pumps the arms to vacuum. LISA would operate in open space.)
I could probably go on, but this may give the idea that what you're suggesting is much more complicated than perhaps it seemed at first.