Don't believe that there is a corporate centurion card, only personal and business.
As to FHR hotels, you are correct that if booked through a Centurion agent, that you will obtain a better offer than if booked on FHR website. That has been my experience in the twenty years that I have had the card. Most recently a Paris trip with stay at Le Bristol where Agent's rate was better than FHR (or Le Bristol's) rate by over €250 per day.