The thickness of the shims is based solely on the gear pattern, which is determined by the spacing of the pinion, relative to the ring gear. The only way to know what size you need is to print the gear pattern.
FWIW, this isn't done on the assembly line this way. The way it's done by the manufacturers is to measure the contact point (the theoretical pitch line where the two gears contact each other) on both the ring and pinion and then calculate the shim thickness required. A sample differential is printed regularly, in order to validate the measuring tools.
You can set up the gears this way, if you have the dimensions of the gears, including the distance from the mounting faces to the pitch lines, and a means of measuring the internal mounting locations of the gears, inside the differential carrier. It sounds difficult and involved, but it's really not. All gear manufacturers supply this information, so that their gears can be set up properly. The only real variable (actually, a set of variables) is the internal machining of the differential carrier.
You need to set the casting on a flat table, with the pinion centerline vertical, pinion input side facing down, and measure from the table to the face the pinion backface would sit, when installed, without shims. You have to orient the casting this way because the pinion seat is on the opposite side of the casting from the pinion input side (the side connected to the propeller shaft). You may have an inspection cover on the back side of the differential carrier that mates to a machined face. Do not use this face to measure from. It's tempting, because you can see both faces at the same time, but this isn't the datum used to set the gear centerline; it's uncontrolled as to distance from any datum.
You then need to turn the casting 90° and measure from the machined face on the ring gear side to both the ring gear mounting face (internal to the casting) and the pinion centerline. You can't measure the centerline directly, because it's not a real surface, but you can measure the bottom of the hole by measuring to a round bar placed in the hole (any size is fine) and then measuring the hole diameter. The rest is all addition and subtraction.
This is the way the assembly line measuring tools are designed. Printing is the easier method. It just takes longer.