= Solution
Put $C=\cos(\chi/2)$ and $S=\sin(\chi/2)$. Commutativity of the two <creation operators> and the <binomial theorem> give
$$
|\chi\rangle=\sum_{j=0}^N\sqrt{\binom Nj}\,C^{N-j}S^j|N-j,j\rangle.
$$
Thus
$$
\boxed{P(N_1=j)=\binom Nj(1-p)^{N-j}p^j,\quad p=S^2},
\qquad
\boxed{\overline N_1=Np,\quad\overline N_0=N(1-p)}.
$$
The state is a <fixed-number two-mode condensate> with binomial mode occupations; it is not a product of two independent Poisson-distributed Glauber <coherent states>.
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