No, because CORS can only restrict which origins (scheme, domain, port combinations) are able to access the site's data. But you're not even connecting from a web origin but from localhost and you're trying to defend from all access except by your frontends. For this, you need a shared secret between the server and the frontend.
A further limitation of CORS is that certain requests are allowed even if they are not from an allowed origin.
I assume someone could have a look at the JavaScript on the browser and see hey this must be the secret stored here because it is passed to the server on every request. Then write there XSS attack to use that.
A further limitation of CORS is that certain requests are allowed even if they are not from an allowed origin.
To conclude, you definitely need a secret.