Brandon,
Nice idea you have there and a great question. Personally, I don’t use compost tumblers because they take the compost and remove it from the earth, and the earth both adds to the compost and is immensely helped simply by having the compost sitting there in place.
However, there are certain times and circumstances where a tumbler is totally appropriate. As for the moisture content, I cannot recommend an amount of water to add with browns. What I can recommend is that the compost
should be moist but not wet. Think of a washcloth that had been wrung out, or used
coffee grounds.
Ultimately you have to be the judge of what is best and it may take some trial and error. If you get your tumbler loaded, give it a couple of spins, maybe let it sit for a day and see what the moisture content is like. If it feels dry, maybe add a cup of water. If wet, maybe let some liquid out. I would think that in a couple of days that you will get the idea of what the appropriate moisture level is—moist but not wet. Ultimately you want all materials to have access to both water and air and hence the moist rule applies.
Best of luck and please keep us updated!
Eric