For one thing, there almost can't be anything wrong with gravity. Any gravitational effect large enough to explain the anomaly would noticeably affect the ephemerises of the outer planets. Yes, it is mathematically possible to dream up some function whereby the probes are affected but Pluto and Neptune are not, but that's awfully contrived and poorly motivated.
My expectation is that the probes have acquired a significant electrical charge during their journey. This causes their trajectories to bend slightly as the probes pass through the sun's magnetic field (to say nothing of the local galactic magnetic field). This bending causes the probe to acquire a momentum component that is transverse to its direction of travel, but because its total momentum is roughly constant, its momentum along our line of sight (which is what the Doppler shift measures) is necessarily reduced. We have no way to measure the transverse momentum component.
If this idea has been rejected, I'd like to see a quantitative reason.
Not that I know of. :-)
Now that I read your post, I recall that we've had a thread (or maybe two) on this topic before. But it's been at least a year. Anyway, kinks like this need to get worked out. Anomalies are where the action is.
Since we are on the subject of gravitational anomalies, do you know of a good explanation of the Allais effect? Might the probe anomalies be a manifestation of whatever causes this other gravitational anomaly?
I hesitate to stray from conventional physics regarding gravitation for vaguely related theoretical reasons, but there are some strange chinks in the mathematical armor of conventional physics models at the quantum level that may be showing themselves. Maybe Gravity Probe B will shed some light on this, given the precision of its instruments.
It'd take an accelerator jock to think of this. Kind of reminds me of GC/Mass Spec. Can we check the estimate of the charge by looking at curvature of the flight path, the mass, and the applied B field? (Wink, nudge)