In fact you could just boil the water to steam and vent it out the top via a surface valve assembly - the interior would act like a vacuum thermace flask.
Once the ice freezes again behind the probe it would protect the fibre... perhaps?
Fortunately something like that wouldn't be too difficult to test on Earth - probe recovery might be tricky though.
A nuclear reactor could produce basically no heat while offline, then be switched on and suddenly provide 100s of kW when it gets to wherever it's going. The hard part in space is radiating away the heat, but if you're on an ice world, that's orders of magnitude easier.
The hardest part I'd see would just be getting into the ice; there's not really any "melting" in vacuum. The constant boiling away of the water would keep insulating your heater from the ice. Meters 1 to 20,000 are probably pretty easy.
https://www.walmart.com/c/kp/water-wiggler-toy
So it can "slip down" by a continuous unrolling process!