this question exposes my ignorance of how lungs work, but in a situation where the lungs were in good working order, would carbon dioxide build up in them (to be breathed out)?
Only until the partial pressure of CO2 was in equilibrium between the volume of air in the lungs and the the body tissues. It's like bailing out a boat -- it only works if you empty the bucket periodically.
could you "breathe" a small volume of something inert in and out, and have a CO2 scrubber?
Presumably, if your lungs were working well enough for that to be an option, you could just breathe air.