If you do, you're giving in to the notion that your role is only as a laborer instead of insisting on your rightful place as part of management. That makes you a troublemaker. Don't think you can write yourself onto the board through provisions in medical-staff bylaws, your contract or some other organizational document. Board composition is determined by board bylaws, which are amendable only by the board. Don't insist that a certain percentage of the board be physicians.
Talent is far more likely to be recognized than entitlement.
Don't go behind the CEO's back and appeal directly to a board member who may be your neighbor or golf partner. That makes you a conspirator. That tells the board that this group of physicians is not yet mature enough to understand how to use organizational lines of responsibility and authority. That's a sure way not to have effective physician participation at the board level for years to come. Your best strategy is to practice education and persuasion, not confrontation.