The effect, if caused by a moon (or moons) would be much more (as in several orders of magnitude more) noticeable on the moon; no such effect has been noticed.
As a general point, I have been unable to find anything about this on the NASA or ESA web sites and also nothing in the Monthly Notices of the RAS (the most recent I have is last week's). I wonder where/if Dougherty has published. Until there is more real information on this, we are limited to speculation. The "dynamo" theory in the BBC article is obviously one possibility; as is the notion that the Cassini instruments may have been measuring something subtly different to that measured by the Pioneer/Voyager instruments. Or it may be something else. Interesting!
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Stephen