I will have to get out and actually try my ring buckle setup. However, I'm wondering if what you are trying to do is loosen things up so you can undo the biner to get the webbing from around the tree? I'm thinking that I'll keep the webbing on the rings all the time in storage (as it is now).
Is it difficult to rig the webbing so there is no pull on the biner? The friction of going around the tree is what should hold the weight, not the biner, anyway. I was taught to rig a climbing rope this way, run it multiple times around a tree, then LOOSLY attach the biner to the rope. The biner was just a safety backup, not what was to take the weight of the people on the rope. I am thinking that I'll hang my hammock the same way. That's why I got 13' of webbing one each end, so I can go around most trees at least a couple of times.
If there is no tension on the biner, then there is no need to loosen the rings. Just undo the biner and unwrap the webbing from the tree. What am I missing?