When I first started playing with this climate puzzle I mentioned to a PhD in Thermodynamics that using a true "surface" frame of reference, the problem was simplified. The PhD in Thermodynamics promptly told me that I wasn't using "frame of reference" correctly. Not that I had selected a poor frame of reference, but that frame of reference just didn't matter. My how thermodynamics have progressed since I was in college.
The CO2 portion of climate change should, with the current levels of CO2 produce between 0.2 and 1.2 C degrees of warming based on a 1900 to 1950 baseline. Since 1900, the warming has been roughly 0.7 C degrees with some portion of that warming likely not directly related to CO2 impact. Since there is an environmental lapse rate that varies with altitude and some degree of internal variability, selecting a frame of reference that is not very specific would likely not produce anywhere near the accuracy required to tease out individual impacts. That is my opinion based on limited pre-turn of the century Thermodynamics. If you pick the wrong layers, you get noise.
That limit could be based on total ocean heat content or relative ocean heat content. Since the NH responded differently during the transition phase than the SH, internal transfer of energy to the two hemispheres is a likely contributor to the change. If that is the case, then in the control phase, less energy would be used to equalize the hemispheres and there would be more energy loss to the atmosphere on its way to space. Since the NH was gaining more of the transition energy, it would have a greater atmospheric impact during the control phase. Why?
In the NH there is more land mass and less ocean mass. The average altitude of the air over the NH land mass is around 2000 meters in altitude which would have roughly 20% of the density of sea level air. It take less energy to warm a lower density volume than a higher density volume. The impact of ocean hemispheric equalization would be amplified. The land mass also has a lower thermal capacity than the ocean mass. Since land mass can store less energy than ocean mass, the NH response would be different than the SH response during any phase. With the NH oceans "charged" and the lower land and air thermal capacity limiting the storage potential, more energy would be lost to space during the control phase in the northern hemisphere than in the southern hemisphere. This may or may not trigger another phase change, depending on the energy supplied to the oceans.
There is no magic involved, the energy storage efficiency just changes. There are dozens of different influences than might be negligible in the transition phase that would not be negligible in the control phase. Since the internal energy transfer in all phases depends on both the ocean greenhouse effect and the atmospheric greenhouse effect, you have to consider both. There is no single, linear climate sensitivity.