Just to be clear you can buy a FC ticket whenever you want. But,
Under normal circumstances if you dont qualify for 1st you would have to reimburse the company the difference between the coach ticket and the FC on that flight. I do not know think you will if first is your only option. Call global travel and ask them to book you see what they say. I have bought a last minute ticket several time in FC but not had to pay because it was the only seat available.