I use 1/8” shock cord to tether a lot of things to my gear strap in the tree so I don’t drop them to the ground. I’m not tethering my weapon, but I do use an M4 single-point attachment sling to keep my crossbow, rifle, or shotgun attached to me while I’m in the tree. You can tie paracord around the narrowest part of the rear stock to create a paracord attachment point for the single point sling, and if you’re concerned about the metal sling clip scratching your weapon, just wrap the area with a generous amount of camoform tape before tying on the paracord attachment point and clipping in. Works great and you don’t have to worry about dropping your weapon if you nod off while you’re tethered in the tree.
It is an adjustment to get used to maneuvering your weapon on a single point sling in the presence of your bridge, but you’ll adapt and figure it out with some time and practice.