Uniform sampling of intersection area of two disks

Given 2D uniform variable we can generate a uniform distribution in a unit-disk as discussed here.

My problem is similar in that i wish to uniformly sample the intersection area of two intersecting disks where one disk is always the unit-disk and the other can be freely moved and resized like here

I was trying to split the area into two regions (as depicted above) and sample each region individual based on the respected disk. My approach is based on uniform disk algorithm cited above. To sample the first region right of the center line I would restrict theta to be within the two intersection points. Next r would need to be projected based on that theta such that the points are pushed in the area between our mid line and the radius of the disk. The python sample code can be found here.

u = unifrom2D()
A;B; // Intersection points
for p in allPoints
    theta = u.x * (getTheta(A) - getTheta(B)) + getTheta(B)
    r = sqrt(u.y + (1- u.y)*length2(lineIntersection(theta)))  
    p = (r * cos(theta), r * sin(theta))

However this approach is rather expensive and further fails to preserve uniformity. Just to clarify i do not want to use rejection sampling.