Would mandating the use of CO2 as a refrigerant also somehow economically motivate efforts to capture more CO2 out of the atmosphere, in a kind of virtuous cycle? Or would there be a Cobra Effect in there somewhere? I'm not familiar with the industries that refine and bottle the stuff.
No, neither - the amount of CO2 that'd be needed for use as a refrigerant is insignificant. There's already a lot of CO2 produced as a byproduct of industrial chemical processes like petroleum refining. Some of it is captured for reuse (e.g. in drink carbonation); most of it is released to the atmosphere.