To address the issue of mismatched region names between your JSON and atlas files, you should ensure that the paths used during the export process are consistent. When exporting, if you want to remove the folder names from the region names, you should avoid using the "Flatten Paths" option, as it may not work as expected in your case. Instead, you can manually adjust the paths or use a script to ensure that the paths in your JSON and atlas files match. Additionally, you can consolidate your images into a single folder and adjust the image paths in your project to ensure consistency. This approach should help in aligning the region names in your JSON and atlas files for successful on-demand loading.

Bugsy Yes, Spinebot mentioned that in the middle of the paragraph:
Additionally, you can consolidate your images into a single folder and adjust the image paths in your project to ensure consistency. This approach should help in aligning the region names in your JSON and atlas files for successful on-demand loading.

Admittedly the text is a bit vague and likely only understood when you already know this answer. Perhaps this recent posting helps which answers the same question:
https://esotericsoftware.com/forum/d/28961-hi-colored-emission-maps/9

when attempting to export spine for on-demand loading with each skin having its own sprite atlas.

@Bugsy Also note that you don't necessarily need to have a separate atlas for each skin. It's sufficient to have separate atlas pages for each skin (also done via packing by folder structure), which results in a single .atlas.txt file but multiple material and .png texture assets, which each can then be loaded individually on demand.