Newman's lemma for distributional communication complexity
There's nothing wrong in your proof, but you can do even better; by taking the average in

$$
\mathbb{E}_r \mathbb{E}_{\mu} \mathbf{1}_{\Pi(x,y; r) \neq f(x,y)} \leq \varepsilon
$$

you can conclude that there's a single $r_0$ for which

$$
\mathbb{E}_{\mu} \mathbf{1}_{\Pi(x,y; r_0) \neq f(x,y)} \leq \varepsilon.
$$

So in particular, randomized distributional communication complexity is the same as deterministic distributional communication complexity, which explains why it didn't get much attention in the literature...