Partially you're right about POST requirements. But it is only used to initialize Oauth flow. According to the documentation:
1) You must make a POST request to obtain a request token from Pocket auth service
2) Redirect user to the auth page: https://getpocket.com/auth/authorize?request_token=YOUR_REQUEST_TOKEN&redirect_uri=YOUR_REDIRECT_URI
Which means that you have to make a simple XHR to retrieve request token and then you can use chrome.identity.launchWebAuthFlow function to begin Oauth flow.