Behaviour was confirmed for certain compound offspring distributions in Equation (7.55) of arxiv.org/abs/1612.02580 The general case would still be highly interesting though.

The problem may also be reformulated in more probabilistic terms: Take independent copies $\xi_1, \ldots, \xi_n$ of $\xi$ and set $$(Y_i)_{1 \le i \le n} = ( (\xi_i)_{1 \le i \le n} \mid \xi_1 + \ldots + \xi_n = n-1).$$ For each $k$, let $N_k$ denote the number of indices $1 \le i \le n$ with $Y_i = k$. Does it then hold that $$\lim_{n \to \infty} \mathbb{E}[n^{-1} \sum_{k \equiv a \mod d} k N_k] = \mathbb{P}(\hat{\xi} \in a + d \mathbb{Z})/\mathbb{E}[\xi]?$$