A quick update. I belatedly tried a tweak that I obviously should have applied long ago – vibration control under the amp. I got a very good deal on four Stillpoints, which, after plenty of research, I chose from amongst the various brand options. Wowsers! Not only the bass, but they greatly enhanced the A200's overall performance, and, by extension, that of the speakers.
I won't go as far as to say that the issue that we have been discussing is fully resolved, but I am now much closer to hearing the full potential of the KIMs, and am very happy indeed.
What this underscores is that likely both the choice of amp, and, at least in some cases, vibration control, can have a major impact. I suppose that the use of tubes in the A200 is why it may be particularly sensitive to vibration.