No, a flash is not required. Although some form of noseband is required. You could also try loosening the flash a bit, and see if he does any better. Also, a drop noseband may do wonders with him. However, if the horse doesn't have a problem-don't fix it. Most nosebands are to fix most of the same things. The horse opening their mouth, trying to get their tongue over the bit, and crossing their jaw. If the horse has none of these problems just use a regular cavesson noseband. Actually, so many people use flashes for dressage these days, but it was originally made for jumping. So that the rider could have a martingale attach to the normal noseband, but then it had the flash for horses that needed it.
"Dressage-The passionate pursuit of perfection by the obsesively imperfect."