Seriously not one recommendation for heated handles. A husky xpg is practically built for ice core samples with its heated carb and handles....
I love a good problem like this. Really get you thinking.
What about a hot knife ( pronounced ka-nife) maybe a rather thin but study piece of stainless heated via conduction, or even a round stainless hot pipe or square that could be melted through the ice while leaving enough solid ice intact for the sample. Power could be from a battery bank or a small generator.
I was thinking that too, but it takes an immense amount of heat to melt ice. Keeping it melted in cold temps is also rather difficult. It would have to be a sleeve, like you say.