Thanks for the suggestion Abby but unfortunately, without including pageWorld.js, the Inbox SDK won't load at all for manifest V3 extensions.
In order to make InboxSDK load, I needed to include pageWorld.js in the build, and copy/paste from the background script.
Denying load of chrome-extension://<extension-id>/pageWorld.js. Resources must be listed in the web_accessible_resources manifest key in order to be loaded by pages outside the extension.
The workaround to get it to function is to include pageWorld.js so that Inbox SDK works. Without it, Inbox SDK will not work.
Including pageWorld.js causes the issue described in this original thread (headers error).
It would be amazing if Inbox SDK could work without needing to include pageWorld.js but it looks to be required for manifest V3.