I assume that most of the heat loss comes from the water itself radiating heat to space, rather than losing the heat to the atmosphere convectively/conductively, right?
I'd put most of the heat loss as evaporation.
It's not monolithic, and you actually have multiple layers emitting, but eventually the loss is governed by the rate at which IR is emitted out to space from the top of the atmosphere.
Loss from the water is three fold - IR/black body radiation from the top layers of ocean, loss via transport from evaporation and re-condensation, and conductive via direct transfer to atmosphere. For sure evaporative loss is huge, but I'm not sure it's the biggest, as it depends on vapor pressure which will be rather low, and that evaporation putting more H2O into the atmosphere actually provides a negative feedback to heat loss (H2O being a fairly efficient GHG).
Once we get a skin of ice, the loss becomes mostly radiative, with a small portion being conductive/convective. The conductive portion diminishes rapidly as the ice thickens and slows down heat flow, thus permitting the ice/air interface to reach equilibrium at a lower temperature.
The presence of ice, even fairly thin ice also slows down transfer internally in the water column. Without open water, we reduce turbulence and redistribution of heat from depth. Without that, the water column thermo and haloclines will stabilize and reduce convective exchange with the surface.