How to design an experiment to quantify back radiation from atmospheric carbon dioxide?
The thing is, we don´t have any instruments(to my knowledge) to measure the intensity of that radiation. But, we measure the intensity of atmospheric emission from space with satellites, which is seen as the "effective temperature". Logically this is the same as the net radiation downwards. And heat transfer then confirms that no net radiation arrives at the surface. Which is equal to "no radiation at all". Because "net" radiation is the only radiation that can heat anything.

There are lots of people claiming that measurements done with IR sensors and pyrgeometers confirm backradiation. This is not true and it is based on a misunderstanding of the function of the device, as well as misapplication of the stefan-boltzmann equation. Both IR-thermometers and pyrgeometers use the s-b equation  in combination with a thermocouple or thermopile(pyrgeometer). They are both a kind of more advanced and more sensitive thermometers. The thermopile consist of a sensor measuring a gradient across itself indirectly by changes in resistance acting on a current through the thermopile. So, by determining the gradient in the thermopile, it can sense the amount of heat transferred to the surroundings using the s-b equation. When someone claims measurements show "back-radiation", they refer to measurements done with these devices. As most interested people understand, a measurement of heat transfer away from the device, cannot show incoming radiation from a colder atmosphere. If you measure something hotter than the temperature of the thermopile, sure, then you have high accuracy. But a measurement of "back-radiation" from the atmosphere is per definition a measurement of something colder than the device. Because the atmosphere is almost always colder(exception is inversion, but that is not relevant). So any measurement of backradiation made from the surface, is a measurement of nothing. It is simply what the device doesn´t measure coming from the atmosphere. This is what I mean with "misapplication" of the s-b equation. It can not, and should not, be used that way. That is, to add flux densities that are not calculated as "net" transfer.

In regards to your suggested experiment. Unless the gas is hotter than the instruments(if using IR-instruments), you will measure a heat sink, not any backradiation. And, yes, that is what greenhouse gases really are. Heat sinks. The earth emit its heat to the ultimate bottomless heat sink in space, the only observed infinity, and greenhouse gases are added heat sinks along the highway for heat, that leads to 3 Kelvin space.