The curve \(z(1+i)+\bar{z}(1-i)=4\) simplifies to \(x-y=2\) (a line). The region \(|z-3|\leq1\) is a disk centered at \((3,0)\) with radius 1. The line \(x-y=2\) passes through the disk, cutting off a circular segment. The distance from center \((3,0)\) to line is \(\frac{|3-0-2|}{\sqrt{2}}=\frac{1}{\sqrt{2}}\). Computing the segment areas gives \(|\alpha-\beta|=1+\frac{\pi}{4}\).