Oh yeah, if you can't get your anchor up, you just paddle upwind/upcurrent until the ring slides to the bottom of the anchor, and, wallah!!!!. I've never tried this with a Bruce claw, but it works with Danforths.
Danforth anchors have a ring which slide up and down on the anchor. Your description of creating your own sliding ring on an anchor is similar to the built in function of the Danforth sliding ring. The 1K Bruce claw does not have big holes to run rope through and tie knots for this sliding ring. I guess you could use 2 clevis pins for this purpose.
With a Bruce claw and ziptie/clevis pin setup, you can pull on the anchor rope until you're almost vertical above the anchor, and then yank to break the zipties. Once you break the zipties, you are immediately pulling from the clevis without having to move upstream of the snag.
With a Danforth you need to put yourself upstream of the snag to get the ring to move to the other end of the anchor. When there is current and wind, paddling upstream in a kayak, while negotiating the anchor rope to move the ring, is not an easy thing to do. I also have a Danforth anchor but I've never had it stuck before. I hope I never get it stuck with a Danforth while in a heavy current.