Thick straps such as the old Binda type (laminated, with outer layer being synthetic) are (for me) difficult to install, but they stay in place nicely, no twist needed. You might try wrapping the strap with a thin strip of athletic tape on each side where it passes through the pedal casting. If you build up the thickness enough, then it can't pass back through. To remove, unwrap the side nearer the non-buckle end.