Shrinkage in drying to 6% MC is about 7.4% on the average. A few pieces will shrink as much as 8% or 0.12". So, the green size needs to be 1.62" so that the final size is 1.5", but most pieces will be thicker. Then you also need to add a planer allowance of no less than 0.06", and a bit more of the lumber is rougher than normal. Than add a little more for cupping warp. River birch does indeed cup more than many species, so 0.12" is none too much. Then because the mill will cut plus or minus a little bit, you need to add 0.05" or more. So, to achieve all pieces at 1.50" after drying and planing, the green average size, also called target size, needs to be 1.50+0.12+0.06+0.12+0.05 = 1.85"
Now, many mills will cut a little thinner, because this assumes that you want every piece thick enough (1.50" after kiln drying and planing), and it is expensive to be perfect. So, maybe 1.80” target is more reasonable. Note that if you were to buy kiln dried lumber, the assumption or industry practice is to have 1.50" or thicker prior to drying for 6/4. This means that at the mill you can forget about shrinkage, planer allowance, and cupping, so you only need a target of 1.55" or 1/16" over thickness. However, tradition says 3/32" (0.09") or even 1/8” (0.12) over thickness, but this was developed when sawmills used circle saws which needed more planing to be smooth and target setting was not accurately indicated on the mill (more plus and minus thicknesses). Today, surfaces are smoother because saws are better and oftentimes the setting on the mill is very accurate so lumber thickness varies only a small amount.