Lewis Little replied to this question by email. I reproduce his reply here:
In the book I have considered only the special theory of relativity. The special theory predicts no change to the actual rate of clocks.
The general theory [of relativity] does predict a change, and this is what is alleged to be observed in the experiments you mention. I say "alleged" because I have not as yet worked out the TEW picture of gravity/general relativity.
However, given the fact that gravity is a real physical effect, not merely the result of a change to one's frame of reference, a change to the rate of clocks would not necessarily be unexpected.
In my own follow-up question I asked Dr Little: What if there were two atomic clocks, stationary to each other and synchronized. Now one clock flies off, moves in a circle, and again meets up in the same spot with the other clock?
Dr Little explained to me (on the phone) that, as this scenario involves acceleration (rather than steady speed/velocity), it would also be part of a General Theory of Relativity explanation - rather than a Special Theory of Relativity explanation - and his comment (above) applies.