Helmets DO NOT make you safer when riding. They do not prevent the chance of a spook or fall. They are simply risk mitigation, they ONLY work if your horse spooks and then you fall off.
Safety measures include: riding within your ability, knowing your limits, desensitizing your horse to scary objects and situations and taking lessons to improve your riding skills.
Please recognize the difference between a safety measure and a risk mitigation measure!!
If you want to show in a helmet, then show in one. When the horse is more experienced, and more likely to win or place, then switch to a hat. It is likely his spazzing will speak for itself and regardless of your headgear, you may not place this time. Go for experience now, and winning later. Good luck!
They say money doesn't buy happiness -- well happiness doesn't buy horses!