I moved the <script type="text/javascript" src="/_ah/channel/jsapi"></script>
tag from main-stage.html
(html for an internal web component) to index.html
(root html for the website), and the CORS error went away.
The <script>
tag was originally placed inside a <polymer-element>
tag, and was (probably) translated to an XMLHttpRequest by Polymer.dart, causing the error.