I don't have the SpinTires Mod because I don't have the patience to disentangle two undocumented programs at once. Since I've never used that mod, I can tell you authoritatively that you don't need it in order to include American Wilds models.
I'm not sure what you mean that you "included the class files". The DLC class files are already present in the game (assuming that you've been keeping it updated) and in the editor (assuming that you've copied or recopied the media files after updating the game). If you include the class files again in your mod directory, this might be what is confusing the game.
BTW, when you publish your mod to the Steam Workshop, the editor automatically randomizes the names of all your custom assets in your mod. This is handy because it ensures that your custom models don't collide with someone else's, even if they started with the same name. However, it means that a bridge_road_01 in your mod becomes 8pshnf34, which then won't work if you don't have all the files needed to make 8pshnf34 work (class XML, mesh XML, .x mesh, and textures).
Also BTW, the occlusion/shadow is computed and baked into the map by the editor, so I'm not surprised that it's still present even if the model isn't working.
I'm not sure why it happens to work for you when using a reference. If you're using one of the built-in reference maps, perhaps the editor recognizes that it's outside your mod directory, and therefore it shouldn't randomize the model names used by the reference.