I think that kind of stratification is only measurable on a much larger scale. Much like the Coriolis effect has no measurable effect on a toilet boil.
It's true that co2 is heavier than air, but the atmospheric pressure differentials between the top and bottom of your room aren't large enough to create any measurable effect. At least, that's my understanding of it. Someone correct me if I'm wrong.
https://www.reddit.com/r/askscience...2_is_heavier_than_air_why_doesnt_it_all_sink/.
I'm working off of the knowledge of high school chemistry from 31 years ago, so you are most likely correct.