Hi Erik,
hmm, I think somewhere between the relatively low coverage and the almost total lack of features lies the reason for your difficulties.
As to why the “other software” 
does a (seemingly?) much better job is up to speculation. I wished many times that some DEV would weigh in and shed some light on that so that people like us, who want to know WHY things happen (and not just how to work around problems) can understand RC better. My personal guess is that RC only aligns fewer images (in difficult sets, which I would say yours are) because it is aiming for a much higher accuracy which is neccessary for the higher LOD in the mesh. The “other software” is probably more forgiving at the cost of detail.
Something you might want to try is a textured background (distinctly different for each orientation of the object - no masks), then each should align well. Once you got the individual components, run an alignment with “align components only” set to true and hopefully the features on the object will be enough to pull everything together.
Cropping might be another way to go, although some other difficulties might arise from that. If, then crop all images in the same way.
Another thought: have you used lens grouping yet? That could just be the solution, since RC uses the info of all images to calculate one single distortion model. In theory, most parts of the frame should be covered somewhere and therefore giving you a better result. If you did not change any setting like aperture or focus in the whole session, then you are good to go. If you did change something, you might want to ungroup the images for a second alignment. Since RC uses the present components, it will very likely only adjust the camera orientation slightly.
And finally, did you try control points yet? Sometimes it can just make the difference. Start with one, somethimes that’s enough already. Be careful not to get trapped in the cycle of “just one more” though… 
Good luck!