As with most mechanical issues on a bicycle, one starts with a knowledge of what needs to happen for the problem to occur. In this case, "ghost shifting" happens only in the small chainwheel would be due to one of three circumstances:

1. The derailleur position or too long a chain causes chain/cassette interference
2. The derailleur moves from it's position relative to the cassette.
3. The derailleur cage is not properly aligned to the cassette (parallel in both planes).

If the problem occurs in other than the largest rear cog #1 is unlikely, but confirming correct B tension adjustment and chain length should be done anyway.

Given the fact that the problem does not occur in the higher torque gears on the large chainwheel we can eliminate #2 without further consideration of the frame involved, shift lever, etc.

Therefore the derailleur must not be aligned to the cassette. But why then does it not occur in the large chainwheel? There are two planes in which the derailleur can be out of alignment. Many people are familiar with the one that involves the bottom of the cage being closer to the cassette than the top, usually occurring from a fall on the right side of the bike bending in the hanger.

The other plane, though is whether the pulley when viewed from the top is not parallel to the vertical front-to-back centerline of the bike. That misalignment most often occurs with a foreign object or accident interfering with the chain or derailleur and causing severe stress. Both of course can occur at once.

The key to the difference in shifting between operation on the large and smallest chainwheel is the rotation of the pulley. If the derailleur hanger is merely bent inward the relative position of the top pulley under a cassette cog does not change (or at least very little). However, if the pulley assembly/derailleur or the hanger is misaligned in the other plane (twisted) the top pulley position will change significantly.

One can easily determine the latter problem by placing the derailleur on the small chainwheel and a middle or smaller rear cog. Pull the derailleur backward to rotate the cage and observe whether the cage orientation changes. You can also check be adjusting the tension to see if that eliminates the problem.

The fix is as follows:
First, align the derailleur hanger - with a tool, not by eye. Then check the derailleur. If you can't see a problem rotating the pulley, test ride. If the pulley seems to be twisting, or if the test ride fails, replace the derailleur,.

p.s. Also, just noticed the "shakedown" reference. Exactly what does that mean - new bike, first hard ride, just installed new components, etc? We are also missing one critical piece of info that one should always include. When did the problem start and were there any other changes or incidents that occurred at or just before that time?