I’m not sure I understand your setup. I thought at first that you had a folder of notes which each contain 1 tasks, but then you mention archiving all the tasks in a note. And I don’t know what your are using Task Archiver for because it sounds to me like it doesn’t move files?
If you can find a way to move files based on a search or a list of links, you can identify all the notes which contain completed tasks and no uncompleted tasks with this search (where Task folder is the name of your folder of tasks): path:"Task folder" "- [x]" -"- [ ]". At the least, you could work your way thru the list manually moving the files.
There is an Auto Note Mover plugin, but it only accepts tags and note titles as triggers. I suppose you could set Task Archiver to use a heading that includes a tag, and use the tag to trigger Auto Note Mover.
Outside of Obsidian you could use a shell script to find and move the files.