'Shake' in final composition, but not within pre-comp?

If you are using Motion Stabilize and you have a really good decent sized area of detail, or better yet 2 of them if you have any rotation in your footage, then a single pass with Motion Stabilize and cropping to fix the edge problem should be all you need. Running a second Stabilize Motion is not going to give you any better results. The way to fix tracking errors is to manually adjust frames or more often than adjusting, just delete a couple of keyframes.

If you insist on running Motion Stabilize twice there is really no need to nest the second comp in a new one. You can just add your additional elements to the same comp. Also, turning on Collapse Transformations may lead to some odd behavior. I would have to check it out.

The biggest problem with stabilizing time lapse happens when there are inconsistencies in exposure and detail level between shots. The default detail area, the inner box, is way too small for most tracking jobs. That may be part of your problem. I've never not been able to get a perfectly motion stabilized shot in one pass with a single pass. I've had to make some adjustments, but perfection is pretty easy to achieve if you have a significant amount of consistent detail in a shot. The default detail area is way too small for most footage.

A screenshot with the modified properties of your layers showing would help. Just press uu, PrintScreen and Paste to the forum or drag a screenshot from it's folder.