I always do this on a night that the weather will be in the 30's overnight. Depending on how hot you get it on how long the cheese can stay in the smoke. With the set up you will be using I bet you could get at least a 6 six hour run. Use a very mild flavor wood, any of your fruit wood chips would be ideal for this. Make sure they are presoaked so as to not have flame up issues. You want just a very small amount of smoke to wisp over the cheese not a billowing smoke.
If you find you like smoke cheese I got myself an a-maze-n-smoker that does this same cold smoke principle, I really love it for this task.
Total agree with keeping the temp in the 50 degree range. Also I slice my blocks of cheese into smaller ones to add more surface space to come into contact with the cheese.
Weber Genesis EP-310, offset BYC, Maverick ET-732, red Thermapen