Partly answering my own question if anybody's interested. Axios doesn't work with POST and PHP - it requires some trickery (it's sort of documented).
Anyway, the FormData method that I've always used before with Laravel projects breaks horribly with Moodle. However, the QueryParams method works fine.
Anyway, the FormData method that I've always used before with Laravel projects breaks horribly with Moodle. However, the QueryParams method works fine.