Can anyone tell me how to search for government fare availability on flights?

I am required to book through our travel agency and so I usually check seatcounter.com before calling them so that I know which flights have availability in which fare classes. But I don't know how to tell specifically which fare class the government fare comes from. I've always been succesful in the past when I saw availability in Y and the next 2 or 3 classes below it, but this week I tried to book a DL flight SNA-PHL that had seats available in all but the lower 4 classes (T,U,L & K) and the travel agent informed me that for this route the Gov fare is in K class. So I learned that apparently the Gov fare comes from different classes for different routes?

So all that said, does anyone know how to find out what fare class a Gov fare is in for a specific city pair?