The problem is that the fonts are temporarily installed. There may also be an issue with the permissions in the fonts themselves, which may not allow copying, but I'm betting that the font manager is the source of the missing fonts.
Install the fonts for real, then package the document.
You can also place the font files in the Document Fonts folder yourself. There's nothing magical about that folder. When InDesign opens the INDD file, it will activate any fonts it finds there if they are called for by the document.