Lawyers with sympathetic clients need to have a media strategy, in those situations where a payer is failing to do the right thing. Start with the premise that every client who has a real, pressing need for medical care ought to have access to it. Identify the payer who has at least an arguable legal responsibility, and a clear moral responsibility, to provide such care. Institutions are not monolithic. For example, in every insurance company, there is some knowledgeable employee waiting to be helpful.
Timing can be an important factor.
Once you have made contact with the company's lawyer, direct access to others becomes problematic. As long as your inquiry is generalized, such that you are not actually advocating on behalf of your client, or trying to bypass already established legal channels, such contact is defensible and desirable.