I really need to make a video that explains this, and have it somewhere people can find easily. It's probably the single most-popular question I get.
External models (such as attached clothes) will not follow jigglebones until they are baked. But, as you've already discovered, baking procedural bones disables the ability to build new movement. And you may have discovered a quirk of how baking them under certain circumstances will remove all motion altogether. Getting this process to work correctly is simple, but it is not at all intuitive - the crux of the problem is the fact that the procedural-bake function is unfinished, and as such is sensitive and prone to misfiring.
Here is the general process:
1.) Spawn your base model and attach outfits to it. It will clip. This is fine. Ignore it. Turn the outfit off if it distracts you.
2.) Make all of your animation to completion. Once you continue from this stage, no new jiggle motion will be added. So make sure you're all done moving stuff.
3.) Restart SFM. I don't care if you've only done one thing since you loaded SFM. Restart it. This goes back to baking procedural bones being an unfinished feature.
4.) Go to the Shot Editor. Do not work in Motion or Graph. Right-click the Viewport, Render Settings. Turn off all the settings (Depth of Field, Motion Blur, Subpixel Jitter AA, Ambient Occlusion). Turn off lighting. Left-click your base model (do NOT right-click yet!). Wait until the FPS counter in the bottom-right corner is as high as it will go (for me, that is 300 / 300; this number varies from person to person).
5.) Once your FPS is maxed out, bake Procedural Bones on your base model. Your SFM will likely freeze for anywhere from a second to a good half-minute. This means it is working. If a bake happens immediately, it's likely it didn't work. Verify this by expanding Unknown, selecting a jigglebone, going to the Motion Editor, and seeing if it has any animation. If it does, you're good to go. If it does not, close SFM without saving and start again. Usually, if you follow all these steps, you'll get it first try, though.
6.) That's it! Turn the outfit back on if you hid it while working. It should now properly follow the jigglebones.