For this achievement you can’t build the “Bath Houses” or the “Ventilation Plant”. You can still pass the law for bath houses if you want to get either Crowded Quarters or Comfortable Quarters (which I recommend) but you will have to break your promise to the people since you can’t actually build it.
And yes without the Ventilation plants the gas will raise the danger in the work areas and there isn’t anything you can do about it (the gas levels do raise and fall randomly each day based on difficulty). So if you want to get this you’ll have to do it on another play-through because people will die and prevent you from getting the no deaths achievement “Bonus Pater Familias”.
Also I’m like 99% sure you can build and use Heaters and Braziers if you need to keep going through winter. I completed the generator before then and didn’t need either so I’m not certain. However you need coal to make the Steel Composites so there’s no way to complete the dlc without coal, and I’m pretty sure heaters and braziers don’t count as buildings.