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Seems the solution to reducing the flutter is to increase the amount of friction to the vertical pivot axis.
Can you Stop Castor Wheel Flutter?

Caster wheel flutter is not caused by the geometry of your caster wheel setup. flutter is a self exciting oscillation caused by the tread contacting the ground flexing in such a way as to force the wheel to go back towards the centered position with more force than it left. Since the tread takes a certain amount of time to flex back there is a lag in the cessation of the force that re-centers the caster wheel causing the wheel to swing past center in an overcorrection to the other side. And then it repeats itself in the other direction. And back and forth. The solution is to dampen out the oscillation before it diverges with some force(friction) resisting the quick oscillation. Most wheelchairs have a way to tighten down the stem to provide more friction. Often wheelchairs with an inexpensive split ball bearing race used on their caster stems will loose their factory stem preload as their bearings wear out providing more clearance and then the stems and forks then spin freely with almost no friction. You want the fork to turn about a revolution to a revolution and a half when you flip it as fast as you can. if it spins around more freely than that you will not have enough friction to dampen out the flutter as your wheelchair picks up speed. By tightening up your stem assembly you push the speed necessary to overcome the friction present in your stem assembly above the range where you will operate your wheelchair. A loose grocery cart wheel will flutter at a very low speed. As you tighten up the bearings the speed required to cause flutter goes up. If you tighten up your caster stem bearing assembly until you get enough friction the flutter will go away.