For the stabilizing it sounds like you need something like
NewBlue Stabilizer or Lock & Load X to batch everything right in the NLE. Check out their tutorial section and see if it’ll do what you need.
NewBlue Stabilizer
https://www.newbluefx.com/product/stabilizer
Coremelt – Lock & Load X
https://www.coremelt.com/products/lock-and-load-x.html
Mocha should have worked in after effects though. In case you still plan to run all your clips through After Effects you can do the following to batch your renders.
1. Import all your clips and drag them to the 3rd icon from the left at the bottom of your project panel (Create a new composition)
2. In the window that pops up you can choose to put all the clips into one composition and sequence them (render them out as one long video), or create a new comp for each clip (maintain the current file structure).
3. Once you’re ready to render.. Select one of the compositions in the project panel and go to Composition>Add to render queue
4. In the new panel click on the down arrow next to the yellow word ‘Lossless’, go to ‘Make Template’. In the popup window setup the render settings you want (edit button), name it and then make it the default video template.
5. In the render queue panel click on the yellow text to the right of ‘Output To:’ and select a common folder where all the files will be rendered to.
6. Now you can select all of the other Compositions at once in the project panel and add them to the Render Queue.
Render.
Good luck,
Mike Sevigny