Had to throw in my opinion here as I own a Bradley.
I think you did the right thing, Bradley pucks are expensive and it is a drag not being able to use your own wood.
Also, when I smoke cheese, I find the Bradley puts out too much smoke. I use something that gives a much more subtle, gentle smoke flavour. It generates virtually no heat at all but is burning real wood.
I use a Pro Q Cold Smoke Generator. You may not be able to get these easily in the US but there is another company that makes a similar product in the US called A-Maze-N (as good, if not better).
You should check these out, I use mine inside my Bradley cabinet for cold smoking all sorts (mainly salmon). With any kind of cold smoking, you get a much better flavour via a little smoke for a long time than lot of smoke for a short time.
For comparison, if I was to smoke my salmon with the Bradley, more than 2 hours of smoke would be too much and leave it bitter and uneven. With the Pro Q, I go for about 12 hours and the flavour is subtle and balanced.
Hope this helps.