I have a 5000btu AC inside my cab. I can barely hear it run while I am at my desk in the garage 8ft from it. No more noise than a inline fan.
You need an AC for that little furnace, no doubt about it.
Your setup looks fine to me, but you could think about doing it up right in there, and forget about the boxes. You are robbing yourself of precious space.
If you do use the boxes, make sure you have the airflow inside of them proper,(correct sized intakes, sufficient cfm exhaust, etc) or your AC is going to be a waste. If the boxes are squared away proper, then the air from the AC room will work fine to feed the boxes.
Good Luck!
edit:
Also...you may want to try and find a manual control AC unit. Usually only the very smallest cheapos are like that, but they are far more reliable than any digital controlled unit.
One momentary loss of electric will cause a digital control to loose where it was and it will default to the off position. While a manual control unit will simply start running again when the power kicks back on.
A small unit is plenty enough for your grow area.
Got news from the accountant today that my tax bill is going to be way less so I've got some money to spend. I'm going to have a large 15' X 30' storage building built in the back and split it down the middle. Half will be for storage and half for my garden Gotta make new plans now for the setup.