Has the arcsine law been generalised to higher order divisor functions?

Well here's the answer for anyone(else) who's interested. We have $$ \frac{1}{x}\sum_{n\leq x}\left(\frac{1}{d_k(n)}\sum_{\substack{q|n\\q\leq n^A}} d_{k-1}(q)\right)\sim \frac{\sin\pi/k}{\pi}\int_{0}^{A}t^{-\frac{1}{k}}(1-t)^{\frac{1}{k}-1}dt,$$
which is the generalised arcsine distribution function, in other words, the $B(1/k,1-1/k)$ distribution.

I'm not inclined to write out the proof in details right now (especially because I asked the question), but it follows along elementary lines using any of a number of theorems on mean values of multiplicative functions (of the Selberg-Delange type for example), partial summation and some basic combinatorics.

There's an extension of the Deshouillers-Dress-Tenenbaum theorem by Bareikis and Manstavičius, http://doai.io/10.4064/aa126-2-5, which almost treats the question you ask. I'm not sure why the condition $f(p^\ell) \ll 1$ is imposed there; as far as the method goes the condition $f(p^\ell) \ll \ell^C$ ($C$ absolute) should be fine.