I am well aware of the external contractor dynamic. I work at a small company that provides services to $Fortune_500_company. My access badge is a different color than $Fortune_500_company employees, and I am not allowed to partake in certain employee perks. In fact, $Fortune_500_company is very clear that despite working on their computer, having their email address, having an assigned seat at their campus, and being on their phone system, I am not an employee of theirs. I am OK with this.
There are plenty of ways to be screwed in your career. Your contractor/staff status should be considered an unrelated byproduct.