I seem to recall a discussion a few years back about this, but couldn't find it after a quick search. As others have pointed out, anytime the anchor and chain aren't aligned perfectly on the sea bottom, any side load will try to bend the forks of the Kong in a way it wasn't designed for. I believe some folks have used a short piece of chain between the anchor and the swivel rather than attaching the Kong directly to a shackle. Not sure if that solves your twisting problem, probably not.
I know you didn't want to go there, but I did away with my swivel to keep things simple. Sometimes any forward way of the boat tends to turn the anchor backwards as the anchor is just breaking the surface. If so I get the anchor clear of the water, then lift and twist the chain a couple times to get the anchor turned properly. A nuisance for sure, but it works. (You could possibly drag the anchor on the surface and motor backwards briefly to get it turned properly, but that can be problematic too in a lot of situations)
Good luck with it.