Thanks for the comments and suggestions. The image I posted is far and away the best shot I have ever taken of the moon (but of course, that's not saying much...). I was pleased, in the spur of the moment and after a couple of glasses of wine, to remember to switch to spot metering and under-expose a bit!
I intentionally stretched the lens to 300mm just to see what it would do on the D7100 under those conditions -- I'm not surprised you find the 70-200 sharper at 200mm even with a more severe crop than the 70-300 at 300. I hadn't thought about the 105 macro for this, but I might try the same thing with my Sigma 150 f/2.8 macro, either alone or with the matched Sigma 1.4x teleconverter (giving me effectively 210mm @ f/4). I was surprised at how much latitude I had the keep the shutter speed up (as you point out, even more than I took advantage of) -- the full moon is brighter than you think! Also, at 1/1000 or faster, VR is pretty useless and some say it may even be counterproductive. (I didn't think of that either at the time -- my shot with the 70-300 was with VR on.) Another approach would be to haul out the tripod and shoot at base ISO, although at some shutter speed moon movement will put and end to ultimate sharpness.
All of this is firing up my latent case of NAS. I think the new 70-200 f/4 might be somewhere in my future as a glass upgrade from the 70-300. I've never seriously considered either version of the 70-200 f/2.8 because it is just too damn big and heavy -- one of the things that keeps me firmly planted in the DX world is the combined cost, size and weight of quality FX-compatible lenses. The 70-200 f/4, however, offers most of the benefits of its f/2.8 brothers at a size/weight approximately equivalent to the 70-300.
Maybe I'll try some of this stuff in a night or two or, failing that, next month when the full moon comes around again.