I've browned several barrels. The best is to use one of the cold browns. Laurel Hill and Wahkon Bay are good. You live in Arizona and if you have room in your car or van to create a mini oven in the sun, you have an ideal browning oven. Make a small rack to lift the barrel free of the floor. Follow the instructions. Be sure that all oils are removed from the metal. Wear latex gloves. You don't want any fingerprints. Apply several coats of the solution, carding or wiping away light rust between applications. This may make a darker brown (almost black) than using the heat and humidity of your shower as a browning oven. It's not really that much of a pain and you have control of the results.
Here you see the results of the browning of my Harper's Ferry Rifle with Track's browning solution. The Derringer was left in the white.
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The Harper's Ferry was browned in the back of my minivan. It got to about 150 degrees F and really did the job.