First of all I would like to thank all of you for your advice - it really helps to know that there is sombebody out there you can turn to (if the people who are actually supposed to be helping and supporting you are both not avaliable and not interested)

The comformal map argument seems to work. Unfortunately I found out that my reasoning pertaining to the case when a point lies on the flat side of the boundary (a surface ball) doesn't seem to apply. Most approaches I was able to finde in the literature do not deal with the case of a reflecting boundary. The best possible thing I was able to find was in Probabilistic Techniques in Analysis (Richard F. Bass)
The book offers a chapter on lipshitz domains and on page 195 the following result ist presented - $\omega(x,A)=\int_A M(x,y)\omega(x_0,d\omega)$ where $\omega(x,A)=\mathbb{P}^x(X_{\tau} \in A)$ and $A\subset \partial D$ with $D$ bein a lipschitz domain.

**My question thus boils down to the following:**
How do I show the continuity of $\mathbb{P}^x(X_{\tau} \in\partial_a U)$ for $x \in \partial_r U$ ?

I have been battling this problem for weeks now :(