The conformal mapping $z\mapsto \frac{1}{z}$ doesn't map onto the unit disc, but it maps to the unit disc minus the interval $(0,1)\subset\mathbb{R}$. I tried using the fact that the circle is perpendicular to the ray and that they have a common point on their boundary, but I am not sure if one point is enough. I am not sure how to proceed because I am not sure how the boundary of the image below will be mapped to the boundary of the unit disc.
I'm not looking for an entire answer, just a way to deal (or get rid of) with this ray and possibly a technique that would apply to a larger class of problems.
First apply $\sqrt z$, the branch defined on $\mathbb{C}\setminus[0,\infty)$: $\sqrt{|z|i^{i\theta}}=\sqrt{|z|}e^{i\theta/2}$. This will take $D$ into $\{|z|>1,\text{Im}(z)>0\}$. Now apply $(z+1)/(z-1)$. I hope you can follow from here.