With the basics of thermodynamic system in conditional equilibrium with steady state heat flow,it is easier to understand the somewhat complex responses to changes in the system. With a lower sink flux limited by the latent heat of fusion, the upper source flux would be limited by the latent heat of evaporation or the the rate of evaporation, if there is sufficient differences in the energy of the two thermal masses. If the heat of evaporation is not the limit, then the rate of evaporation would allow a range of temperatures for the source.
Carefully consider this special case of conditional equilibrium. You have two or more objects in a steady state which is in equilibrium with another common steady state. One steady state would be controlled by a phase change which maintains a stable temperature while there is sufficient matter to undergo that phase change. That stable temperature maintains the second steady state condition with another object or ambient in this case. Since the energy absorbed or released during phase change is much greater than that required to change temperature without a phase change, there would be a conditional equilibrium.
For a simple example consider the freezing temperature of salt water, ~271.25K degrees. The radiant energy of water at that temperature is 306.9 Wm-2. With common insulation, the energy flow to this sink would be equal to the energy flow to the ambient sink, if the system was in equilibrium. If the total energy flux of the the objects insulated from ambient is 390Wm-2, then the energy flux into the internal sink would be 390-306.9=83.13Wm-2 which would equal the rate of evaporation. For fresh water the freezing point is 273.15K which would radiate 315.6Wm-2. With an average energy under insulation of 390Wm-2, 390-315.6=74.4Wm-2. With salt water freezing and fresh water melting, the rate of evaporation at the source would range from 83.13 to 74.4 Wm-2. Both conditions are much greater than the heat loss or gain without a phase change.
Relative to ambient, imposing a greater resistance to heat loss would increase the internal temperature until a new conditional equilibrium state is obtained. Reducing the loss to ambient would reduce the transfer from the internal source to the internal sink by an equal amount if the system does return to steady state. Since the internal sink temperature is 271.2K for freezing and 273.15K for melting, these temperatures would increase greater than the source. A full 3.7Wm-2 impact at the 271.5K (306.9Wm-2) freezing point would increase that temperature to 272.06K, a 0.81C increase, for a 3.7Wm-2 increase in the resistance to heat flow.
Since the internal system will seek to return to a steady state with ambient, the effective ambient sink would also increase by 3.7Wm-2. 272.06K is still below the freezing point of fresh water, so there would be an increase in atmospheric freezing of water vapor.
Why would this be a valid conditional equilibrium? The "stable" temperature varies. Because energy cannot be created nor destroyed. While the temperature may vary, the energy released during fusion would be the same as that gained during melting. There would be a small shift in temperature requiring a small change in energy, but the net energy of the cycle would remain the same.
Based on this simple model with complex conditions, 3.7Wm-2 of additional "forcing" would produce 0.81C of average temperature increase with an increase in the rate of precipitation. This model does not consider the individual radiant spectrum changes or potential chemical responses to the change in "forcing". The model can be expanded.