You can go to the ends of the bad tape, make a physical splice outside the shell, then re-wind the new tape into the old body. You just have to be sure you won't run that splice past or over the heads. Do you feel... lucky?
Screwless tapes are glued or ultrasonically welded; so, a very sharp razor knife will open it with just a few passes across the seam. Have a camera running to capture what the layout is as you open it, so you know how to piece it back together. Or disassemble a donor tape for practice first.