This is what I am thinking now, which I can do without cutting the stock cord:
I can attach two rings to a biner with a short piece of cord, then use the hammock cord to make the Garda hitch in the rings. To set up I would wrap the tree hugger around the tree, thread one end through the opposite loop to snug the tree hugger against the tree, clip the biner onto the hanging loop and tighten the Garda hitch.
In the event I have to hang from a tree too large for the tree hugger to go completely around, I could simply undo the Garda hitch and run the hammock cord through the two loops and tie the standard Hennessy lashing.
Both of these choices would keep the biner away from the tree and remove any chance of it damaging the tree.
When I look at the choices of cord on the REI site it rates them all in kN. What rating would I need for the cord tying the rings to the biner?