Instead of two layers there are four, deep ocean, surface, ABL and upper free atmosphere. The deep ocean has about 1000 times the heat capacity of the atmosphere so it take a very long time to charge. Right now it is charged to about 330 Wm-2 gaining about 30Wm-2 during the day from ol' Sol and losing about 30Wm-2 in a 24 hour period. The surface gains about 300 Wm-2 on average from ol' Sol during the day and loses about 330 Wm-2 during a 24 hour period, the 300 Wm-2 directly gained from ol' Sol plus the 30 Wm-2 just passing through from the deep ocean. The ABL gains about 130Wm-2 from ol' Sol during the day and loses about 130 Wm-2 during a 24 hour period, 75 Wm-2 of that is the average energy provided by ol' Sol and 55Wm-2 is provided by the surface which has energy provided by the deep oceans. The ABL cannot contain all of the energy available, so a larger amount passes through the ABL. The upper free atmosphere has the least heat capacity, it absorbs about 20 Wm-2 from ol' Sol during the day and releases that plus about 10 Wm-2 is absorbs from the combination of the ABL, surface and deep ocean. The net out of the top of the upper free atmosphere is about 230 Wm-2 which is just an estimate. Depending on the actual energy provided by ol' Sol and that pesky "other" energy (tidal, rotational, internal, transport, waste heat, etc. etc. that is too small to be counted, but adds up to about 4 to 8 Wm-2) plus any energy imbalance which is typically 1 Wm-2 or less.
Each of the four simplified layers interacts with the other layers. If you add CO2 or other stuff to the atmosphere, it will change the interactions with delay, increasing the time constant Tau, being the one most discussed. There are four time constants in this simple illustration and changing one impacts the others. The estimated change per doubling CO2 is about 4Wm-2 which puts it in the "other" energy category. The object of the diagram is to show that ol' Sol has impact going in and going out. CO2 just changes the rate of going out. If more energy is absorbed in the upper free atmosphere, ol' Sol may provide more energy to the upper free atmosphere. Any change in the heat capacity of the atmospheric layers may change in the distribution of the energy provided by ol' Sol to all of the layers.
To add to the fun, ol' Sol tends to change a bit more than was once suspected. The 30 Wm-2 in the deep oceans and the 20 Wm-2 in the upper free atmosphere are provided by UV and visible light with just a dash of near infrared. If the upper free atmosphere absorbs more, the deep oceans would receive less and vice versa. Since Solar impacts in both directions, that change can have twice the impact on either or both of the middle layers. The deep ocean with its huge heat capacity doesn't care how long it takes, but will slowly adjust to whatever the situation in order to restore its preferred balance with the surface.
This brings us back to those poorly understood Solar variations. If you estimate just Total Solar Insolation (TSI), the change is not enough to worry about. However, if you consider the change in Solar spectrum irradiance and the impact on the upper and lower layers, there is about 4 times the impact. Just compare the Lean 2000 TSI reconstruction,
That is a big difference between what actually happens at the surface versus what is happening at the top of the atmosphere.
Added for Webster: