I don't get what you're doing with the paper.json here
It is that simple: if you the paper.json is the item you use for ModelEngine (which seems to be the case) you take the latest generated one and overwrite any existing other in your resource pack. Don't start tweaking it or whatever, you're gonna break everything, as you don't know how ModelEngine will fetch the custom model data.
Also, on your video, it seems that you're adding existing stuff from littleroom to the paper.json, without respecting formatting constraints in the JSON file, nor ordering of custom model data.
My recommendation would be to use another item than paper.json, as you don't want to have to handle conflicts for ModelEngine files. You can pick something like TIPPED_ARROW instead for example.
Finally, I've seen you had a lot of models, so make sure to put them in subfolders or you'll get a partitioning issue in ModelEngine, that can lead to similar issues