I live in Tucson; I know exactly what you are experiencing.
I used many methods until I built a CO2 system.
I saw tiny tanks featured in an ad in Precision Shooting years ago. Knew a guy who tried one. He told me the tank was too small to be any good. More tank volume is required to get the cooling effect of the expanding gas. The smallest tank you can use and still get the cooling effect is a 20 lb tank as seen in the picture.
The picture is old and shows a regulator. Unless you already own a regulator, you don't need one. I found out later all that is needed is a ball valve. There are a few brass fittings which the welding supply company should have on hand.
A rubber hose is attached and the tip is tapered so it will wedge into the end of the chamber.It only takes a few seconds to feel the barrel begin cooling. I stop when it gets close to ambient because the cooling will radiate after the valve is turned off.
The way I use the tank is to place hose in chamber, HOLD it in place then open up the valve on the tank then slightly open the regulator or ball valve to get the desired flow and save gas. If the regulator or ball valve should already be open and you open the tank valve without holding the hose you can get some interesting welts on the face, been there done that.
There are at least five guys at our range that use this system from time to time. Even one custom builder, Dave Miller. We can find no ill effects of using this system. I don't use it all the time but when things are hot or I want to work up a load it will shorten range time considerably.
Two days ago I was working with a 257 weatherby with temps in the 90s. I would shoot two then cool for 5-10 seconds, which was enough to cool the barrel to ambient.