You are riding him differently in the show pen than the rest of the time. Go to playdays and practice ropings. Ride hard outside of the arena and when there is a 'dead' period, go stand in the far corner of the arena and 'rest'. At the break when they are putting the cattle back in the pen behind the chute, go drive them back through the return alley for the ropers. Just make sure yo get your butt out of the arena when they are roping. That is when you go out and work somewhere out of the arena.
Every time you work a pattern in a show, leave the arena and work hard in the warn-up area. What most people do is work in the warm-up area before a class, show the class and then go to the barn or trailer and quit. Horses learn very quickly that the quicker they get out of the show pen, the quicker they are done. Always work one harder after a class than before it.
Stop just loping circles; they are too predictable. Start loping squares. On the big outside riding area most reining trainers have, lope squares on the inside lead making nice rounded corners and very straight sides. Then lope in the outside lead making a small very round 3/4 circle to the outside at every corner. This kind of routine keeps horses guessing. To keep them honest, you cannot fall into patterns and routines that they think they know what is coming. They have to keep listening.
But probably the biggest thing you can do is to work harder after coming out of the show pen than you did before going in. Then a horse will not get so anxious to be done and get out of there. It just means more work to them.