I find when getting directly in water below about 35-37 degrees that my exposed face and lips have a sharp burning feeling for a couple minutes until it all goes numb and then I usually notice some minor post dive swelling for an hour or two.I know what you mean. I have more trouble in the beginning of the dive after a few minutes in the water my face gets used to it and all is well. At times my lips have been so numb that I almost have to hold the reg in.
I do not have the same symptoms in situations with thermoclines where the bottom temps may be in the same 35-37 degree range but where you get to and leave those temps progressively.
So on a true ice dive, I'll often use an FFM, otherwise there is just not enough benefit to offset the compromises unless coms are needed.
With a standard regulator in very cold water, an Aqualung comfobite mouthpiece works well as much of the work in keeping the second stage in place is done by the mouthpiece resting against the roof of your mouth rather than by jaw pressure. Also, it is critical that the hose lenght be correct as a too short or too long LP hose wiill pull or push the second stage and require much more jaw pressure. Hose lenght is a much bigger factor in jaw fatigue than the weight of the second stage.