Simple Rack and Pinion Zero Backlash (no springs)

[Muzzer]

Actually you don't need to calculate the tension force. This is because the tooth-to-tooth force is not going through a spring. The belt tensioning force need only be such that facilitate the "touching" of the teeth of the respective pinions in each direction.

If you have no preload tension, then by definition you have some backlash in your system. There are 2 approaches to this - either a spring tensioner to maintain a tight belt / chain and take up any wear or unevenness - or a fixed tensioner. With the latter, you will always end up with some backlash. With the former, you must ensure the preload tension exceeds the maximum drive tension. You seem to have the latter but either means could be used here.

You get the same result with a dual pinion setup if you have an adjustment screw between the pinions, rather than a spring. That's essentially what is used on horizontal milling machines that have a backlash adjustment.