You can put one image in a separate atlas page by putting it in a subfolder:
http://esotericsoftware.com/spine-texture-packer#Folder-structure
It's rare to use 8-bit color (256 color palette) with an atlas. Spine doesn't output an 8-bit atlas, even if you choose to reduce PNG frames down to 8-bit, the atlas will be 32-bit.
How are you converting your atlas to 256 colors? You get a PNG or JPG atlas from Spine, then you convert that using some tool? Which one? I suggest using gifski because it does a good job with the quantization. If you want PNG you could turn the GIF in to 8-bit PNG afterward.
If you export 8-bit PNG or GIF from Spine, the colors will be good within each frame. To combine them you'd have to do quantization again and results will be poor. It'd be better to output 32-bit, pack the atlas, then reduce the colors. This allows the palette to be chosen for all colors in the atlas page, but how that is done and if it is done well depends greatly on the tool reducing the colors.