John,
Very inventive, but your choice of using bowden cables is the downfall.
No matter what you do, you will never get them to have zero backlash unless it it a perfectly straight setup, and how you have done it, tensioned with some sort of spring. The problem is the running clearance between the cable and the outer sheath, and going around corners, you will always get some flexing of the outer sheath.
The method was used many years ago in the beginnings of DRO's on model engineering equipment, but without the outer cable. It operated similar to a steel tape measure, where the amount of cable pulled out, turned a rotary encoder. It did have the advantage that it could be used with a rotary table as well, as the cable just wrapped around the turny bit on top.
I don't know if anyone actually bought any, as they soon disappeared from the market.
I wish you luck on your efforts to get zero backlash.
John