The texture packer that comes with Spine should be deterministic as long as your image dimensions don't change.
For example, if you packed Spineboy's images. And then had an alternate version of those images without changing their size, it would pack with the same layout.
This means you can have the normals generated per image, then pack them into their own png. This allows both the diffuse and bump textures to line up correctly.
I'm not sure how Sprite Illuminator works (if it allows you to easily generate normals for a bunch of parts at the same time.)
If not, you might be better off generating normals on the packed atlas.