I too live in Tucson. I had temp/humidity problems when I first started a few months back. It took just over a month but I got my temps/humidity just right. First off, I have a swamp cooler that I run most of the day. I live in a small one bedroom triplex so I had to find a spot to put the tank so that the flow of the cooler would affect it as little as possible. When the cooler is running it will drop my temps by up to 3 degrees. I have a 40gal glass tank with a dual sliding screen top. I have an under the tank heater and a 100w ceramic heat emitter in a reflector lamp. Both are attached to a Zoo Med Repti Temp 500R Thermostat. The CHE is hung about 1 inch from the screen top on the hot side. I have one regular light to simulate day/night set up on a timer. I figure since BP's are nocturnal they don't really need the UV. I have a towel folded in half over the majority of the top. I use the Zoo Med forrest bed (comes in brick form) for my substrate. I mist twice a day. My temps are 90-95 on the hot side and 82-85 on the cool side and my humidity ranges from 52-63%.
What you could do is get rid of all the lighting and just get a CHE to put over the hot side. They work great and can be run 24/7. If you don't want to do that then I would move the black heat light to the hot side instead. You really don't want anything over the cool side.
I tried many things before settling on what I have now. When I first started I had a simular setup that you have. My temps and humidity were all over the place. I bought the digital temp/humi gauge from walmart and love it. I tried the red basking heat bulb and it blew up after only a week of 24/7. You just kind of have to play with things to see what works. You might also think about moving the tank to an area with less airflow for the a/c or cooler.