If only one in a billion CO2 molecules actually gets to hold on to energy in the form of IR that it absorbs long enough to actually radiate it, of what importance is either your circle, or your sideways escape area?
There are 2.53 10²⁵ molecules of air per m³. If CO2 is 400 ppm, that would mean there are roughly 10²² CO2 molecules per m³.
If only 1 billionth can radiate, that would mean 10¹³ CO2 molecules are radiating a photon every 26 microseconds. (26us is the mean vibration relaxation time.)
In one second 10¹³ / 26*10⁻⁶ = 3.8 10¹⁷ molecules are radiating.
Planck's law gives the energy of one 15 micron photon as 1.32 10⁻²⁰ Joules
The total radiation per cubic meter is 3.8 10¹⁷ x 1.32 10⁻²⁰ = 5 10⁻³ Joules/sec or 5 x 10⁻³ Watts. It's small but much larger than convection.
.