Just a side note (sorry, pet hate of mine), percentages don't apply when talking about degrees celsius (or farenheit for that matter). In reality, the 0 point to refer to for percentages is actually -273C, aka 0 Kelvin. A percentage change in celsius is actually pretty meaningless...
Good catch.

Let's say 318Kelvin (45C) - lower it 10 degrees. 3.14% decrease, right? Not as impressive.

Unless the temperature scale isn't linear, if it's logorithmic, I have no clue whatsoever as to figure out the percentage.
I'm not worried about the buy price for a one room air conditioner, it'd be the electricity bill i'd be worried about.
So venting the heat that comes from my exhaust fan to behind the wall wouldnt work? I don't see why not.
How much does your electricity cost right now per kw/hr?
I've got a small A/C, a few years old; never models might be more efficient. 5.6A draw, and let's assume 650W. And the compressor won't be on all the time either, so actual usage would be less.
So let's assume a high price of 12 cents per kilowatt-hour. That should be 7.8 cents per hour. Assuming you run it 24/7 (possible, don't know what nights are like in AZ), and the compressor is on all the time (not likely, even that far south, especially for a small room), it'd be $56.16 per month. I'd imagine it to be considerably less than that though. And bear in mind the price per kw/hr I used; yours should be a good bit less than that, or so I would hope.
So if the AC is still out of the question, either what someone suggested with a fan to circulate to another room, or to outside, or....well, that exhaust into the wall might work.
Thing is, since the wall is of course adjacent to your room, a section of it is going to heat up (only a section, because of vertical support beams), but the wall is going to warm the air it touches on both sides. Of course, if the adjacent room is cooler, the wall itself might act as a (inefficient) heatsink. But only half of it.
Or else just outright vent it directly into the adjacent room. You'd need a decent fan though, which may be loud.
This fan is a powerful sucker. Loud, but it moves a lot of air and has a lot of power. Do use fan grilles, as I'm sure it is more than capable of drawing decent amounts of blood.
It might have rpm monitoring, but use
an adapter, otherwise you'll fry your motherboard's fan headers.