OK, I somewhat figured it out.
I was able to expose the eyelet of the cable where it attaches to the cross piece.
The problem I was having is that the eyelet of the cable was actually riveted to the corner of the cross piece under the canvas. There is no fastener piercing the surface of the canvas.
I drilled out the rivet, exposed the screw hole in the cross piece, and routed a new cable.
The question I have at this point is whether the hole in the cross piece is threaded to accept the screw (that pierces the canvas and holds the eyelet), or does the screw use a nut. I suppose that I could as well just tap the hole if that is what is required given that there was a rivet there originally..