It's been a while since I worked on the LTI stuff and the LTI interface itself has changed over the last 2 or 3 years. Originally I thought consumer key was an important key for passing stuff back to the LMS but perhaps not. It appears to be set in WeBWorK when WW receives the authorization from the oauth packet along with the request from the LMS.
It is set to the oauth_consumer_key parameter in the request. It is then returned in the grade packet along with the consumer_secret.
Let us know what you find out.