Same issue here. Did you find a fix?
If I add the Basic Auth plugin, and ensure that it adds it’s htaccess rules then the whole wordpress site goes behind basic auth. So visiting whilst being unauthenticated shows the dialog box.
However when I go to use the WP Rest API (v2) it seems to completely bypass the Basic Auth. So my user never gets authenticated and as a result none of the endpoints that require a user work.