My grow space is similar - 5'W x 8.5'L x 6.5'H. I'm still setting this up as a new grow space so it isn't tested yet. I'm planning a 340 CFM 6" inline fan with a big fat carbon scrubber. Now do have intakes and exhausts cut into my walls so that is different. Also, I'm not cooling anything so it is just for ventilation and odor control.

You could just stick the carbon scrubber and fan in there and have the fan scrub the room and return the clean air locally. It will deal with the odor issue assuming the fan is sized correctly. You have AC coming in but you can not count on it to cool your room unless it is running 24/7 or unless the thermostat is in the closet.

Is it an attic above? If so, you could go up to the attic, disconnect and cap the duct from the vent and then use that as your exhaust. That would be ideal. If you can't do that and if you really can't cut any holes then your only real choice will be to vent and scrub to the same place which does not help you with heat. Why no holes? Is it drywall? My kid can patch dry wall. Seriously.

Also, given that you share the grow space with your room, you should look at quiet fans. Some of the inline fans can be pretty damn loud. I'm using a Panasonic Whisperline inline. Best price is from westsidewholesale.com

Hope this is some help.