single post templates for custom post types - for preview purposes

I have custom post types that have no single post view (not necessary for this site). They were created using the Custom Post Type UI plugin and use the Advanced Custom Fields plugin for custom data (lots of it). The front end of the site is fine, but the client would like to be able to click "preview" in the WP back end when he's editing and check what it looks like before he publishes. I thought no problem - created a single-recording.php template (for a post type called "recording") and copied the output code into the single post loop.

When I click Preview from the post edit screen however, I get a 404, not my single page template. Any idea what the missing step is?