Just to note - This actually still does not work when making the request via the Shopify app proxy while on local development. It does work ok when the theme is deployed and running from the Shopify server.
Although making requests to the app URL (with the tunnel running feeding traffic to the app running locally) does work in local development. So at least this makes it possible to develop locally. The app URL can be swapped out for the Shopify app proxy when it comes to deployment. This won’t be a great dev experience if/when the app proxy signature is checked on the endpoint provided by the app, but that’s a problem for another time.