So that at least there’s a solution here, will add that this plugin seems to succesfully migrate Posts as well as category and hierarchy.: https://wordpress.org/plugins/post-type-convertr.
On the production server, I also needed to run the following SQL query after migrating with the Post Type Convertr plugin:
UPDATE wp_term_taxonomy SET taxonomy='project-category' WHERE taxonomy='product_cat';
I dunno. Sorry it’s not more useful information.