It is nowhere near as simple as that.  All the radiation gets to space sooner or later.  Carbon dioxide just scatters it on its way so you don’t see radiation in those bandwidths at TOA.  The energy still gets out, and you have no proof that it doesn’t, because you don’t have the necessary simultaneous measurements made all over the world.

In the hemisphere that is cooling at night there is far more getting out, whereas in the hemisphere in the sunlight there is far more coming in.  This is obvious.
When I placed a wide necked vacuum flask filled with water in the sun yesterday (with the lid off) the temperature of the water rose from 19.5 deg.C at 5:08am to 29.1 deg.C at 1:53pm while the air around it rose from 19.0 to 31.9 deg.C.  

What did the backradiation do at night?  Well from 9:15pm till 12:05am the water cooled from 24.2 deg.C to 23.4 deg.C while the air cooled from 24.2 deg.C to 22.7 deg.C.

According to those energy diagrams the backradiation, even at night, is about half the solar radiation during the day.  Well, maybe it is, but it does not have anything like half the effect on the temperature as you can confirm in your own backyard.

This is because, when radiation from a cooler atmosphere strikes a warmer surface it undergoes “resonant scattering” (sometimes called pseudo-scattering) and this means its energy is not converted to thermal energy.  This is the reason that heat does not transfer from cold to hot.  If it did the universe would go crazy.

When opposing radiation is scattered, its own energy replaces energy which the warmer body would have radiated from its own thermal energy supply. 
You can imagine it as if you are just about to pay for fuel at a gas station when a friend travelling with you offers you cash for the right amount.  It’s quicker and easier for you to just pay with the cash, rather than going through the longer process of using a credit card to pay from your own account.  So it is with radiation.  The warmer body cools more slowly as a result because a ready source of energy from incident radiation is quicker to just “reflect” back into the atmosphere, rather than have to convert its own thermal energy to radiated energy.

The ramifications are this: 

Not all radiation from the atmosphere is the same.  That from cooler regions has less effect.  Also, that with fewer frequencies under its Planck curve has less effect again. 

Each carbon dioxide molecule thus has far less effect than each water vapour molecule because the latter can radiate with more frequencies which “oppose” the frequencies being emitted by the surface, especially the oceans.
 
Furthermore, it is only the radiative cooling process of the surface which is slowed down.  There are other processes like evaporative cooling and diffusion followed by convection which cannot be affected by backradiation, and which will tend to compensate for any slowing of the radiation.

This is why, at night, the water in the flask cools nearly as fast as the air around it.  The net effect on the rate of cooling is totally negligible. 

The backradiation does not affect temperatures anywhere near as much as solar radiation, even though its “W/m^2″ is probably about half as much. 
 
And there are other reasons also why it all balances out and climate follows natural cycles without any anthropogenic effect.  This is explained in detail in my peer-reviewed publication now being further reviewed by dozens of scientists.

http://principia-scientific.org/publications/psi_radiated_energy.pdf

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