Plume from opening lid of box

I’d like to open a chest freezer (in case that’s not very international, Americans often have a freezer in the garage or basement that is very basic and it’s just a deep box with a lid on top, sometimes also called a “deep freezer.”) open. When you open it, You’ll get some fog inside of it, but because you opened the door, some of that air will pull out and spill to the floor.

But that’s where I have difficulty knowing how to do this in mantaflow. Ideally I want to just have a volume of smoke (i’ll just call it that because that’s the fluid type) on frame 1. It shouldn’t really be creating anything. Things in that volume are cold. Things outside of that volume are not cold.

I’ve had zero success setting the box to volume emission. I currently have it set to surface emission at 1.0. I turned on use flow, and turn it off after a few frames. I set the temperature to .5 and the density to 1.1

In order to keep this thing Somewhat reasonable in terms of size of the domain, I should be dissolving the smoke. And I have it looking OK, but the dissolve is on the domain. So since all the smoke existed at once, it all fades off at once.
IS there a way to cause it to dissolve only once it exits?

I Could have two outflows I suppose, where a plane near the bottom of the freezer is just slowly emitting a little at a time, which should keep a certain amount visible? I’m just wondering if there is an efficient way

I also thought I had it looking ok, but that was only eevee, but I can’t get volumes to be occluded by other objects, so would definitely have to get compositing this (which is fine). But in cycles, I wasn’t able to get it to be the right density.