Is the run under/part of the coop the only run you have, or is there an additional larger one? Because if that's what's getting tarped, even partially, I can see that impacting airflow into the coop itself, as there's simply not a lot of air volume in the overall area.
Since I know it's winter it's probably impossible to address right now, but if that's the entirety of your run space, even for four bantams you're likely to run into behavioral issues as they get older. It's simply too tight. Even though bantams are smaller they can be as active as standard sized birds, so I'd still recommend sticking to the minimum of 10 sq ft per bird (or my personal recommendation is to not go smaller than 100 sq ft even with a small flock, both for the birds' well being as well as your ease of maintenance).
In many cases they're designed for chicken, not chickenS. Lack of space (whether floor space or wall space for ventilation) is a big shortcoming of prefabs, unfortunately.
12 sq ft coop (provided that does not include nest space) is probably enough for the birds you have as far as floor space, however you don't have the depth needed to do any form of deep litter (or more properly in this case, deep bedding) in there. Not sure what your clean out schedule is but provided the material is dry and not too soiled or smelly then however you're managing it is working fine.