Simple practical, non-toxic, non-flammable reaction for slow release of CO2
I would recommend not touching or coming near calcium carbide at any rate. The acetylene it creates is explosive in certain concentrations (with a flame to ignite it) and notorious for extremely loud bangs. And it also is rather a diversion for what you want to do.

Instead, consider carbonates. Baking soda is one of the prime household sources of carbonates, and you just need to add an acid to release carbon dioxide; and nicely, the only by-product is water. You should use rather dilute acids so that you can slow down the reaction rate. Vinegar might not be the acid of choice (it has a distinct smell) but either essence of vinegar diluted or citric acid (which you should be able to buy from baking suppliers, too) should work. Or lemon juice.

To make the reaction sufficiently slow, you would need to (as I said) dilute the acid let it be poured into the baking soda solution slowly. Experiment a little! For a heat source, you could likely just put your average heating cushion underneath.

(Personally, though, I would use the steam cleaner because it sounds most kick-arse!)