Correct way to use a refresh_token in OAuth API

Your CURLOPT_POSTFIELDS should look similar to the below:

curl_setopt($ch, CURLOPT_POSTFIELDS, [
    'grant_type' => 'refresh_token',
    'client_id' => $this->config['client_id'],
    'client_secret' => $this->config['client_secret'],
    'refresh_token' => $this->user->refresh_token
]);

You should then get back a response similar to this:

{ "scopes": "some-scopes", "access_token": "new-token", "refresh_token": "new-refresh-token", "expiries_in": 3600 }

I also recommend reviewing these other resources on OAuth: How To Create an OAuth App with the Linode Python API Library
Include required grant_type parameter in docs for refreshing oauth tokens