

You seem to be assuming that the volume is immediately replaced by the external atmosphere, which I doubt is valid
No, I was assuming your volume decreases. I don’t actually know that to be the case, but my assumption is that there isn’t “extra” space inside a person, and so if you lose material from a part of your body that isn’t encased in anything rigid your volume decreases slightly.
So maybe I did have my terminology wrong. When a hot air balloon deflates, it falls. The density went up, but that’s not what’s directly relevant. The weight went down, I guess, but the “number on the scale”, weight minus buoyant force, went way way up, because it lost some lower-density volume that was making the whole thing float. The weight (in a strict physics sense) went down, sure. But the number on the scale (which I was incorrectly calling “weight”) went up. Same thing for a farting person.
It doesn’t completely work that way, just like for humans. Sometimes feeding pets less is just subjecting them to pretty severe discomfort and hunger, while their metabolism is deciding that food is scarce so they better hoard every calorie they can spare. I know it’s significantly urgent to help them lose weight because of the health impacts, but IDK that it is super simple once you’ve decided to try to make it happen.