Your pictures don't show up.
The pictures could still explain your questions better.
Moving a rack a certain distance can be roughly calculated with R(the gears pitch diameter/2)= length of stroke/1.5708 to get the arc length of say 90deg.
Or stated another way,
In your application as I read it. You have known lengths of strokes, but you need to select the nearest acceptable pitch diameter to develope a given rack stroke based on your degrees of rotation for an equivalent arc length. At the same time realize that there can be force gains and losses in the transfer.
You can leave some adjustment in your arm length pivot point linkages to adjust for fine tuning or set stops on the rack to limit travel.
See McMaster choices pg's 1039-1041