= Solution
Use a real, even interaction $V_{-\mathbf k}=V_{\mathbf k}$. Let $\mathcal V=L^3$ and $N_{\rm ex}=\sum_{\mathbf k\ne0}b^\dagger_{\mathbf k}b_{\mathbf k}$. In the <Bogoliubov theory> expansion, keep terms with zero or two noncondensed operators. The zero-<momentum> term gives $V_0N_0(N_0-1)/(2\mathcal V)$. The terms with two noncondensed operators give
$$
\sum_{\mathbf k\ne0}\left[\frac{N_0}{\mathcal V}(V_0+V_{\mathbf k})b^\dagger_{\mathbf k}b_{\mathbf k}+\frac{N_0V_{\mathbf k}}{2\mathcal V}\left(b^\dagger_{\mathbf k}b^\dagger_{-\mathbf k}+b_{-\mathbf k}b_{\mathbf k}\right)\right].
$$
Here $V_0$ is the direct condensate interaction and $V_{\mathbf k}$ is its exchange contribution. At fixed total <particle number>, $N_0=N-N_{\rm ex}$, so
$$
\frac{V_0N_0^2}{2\mathcal V}=\frac{V_0N^2}{2\mathcal V}-nV_0N_{\rm ex}+O(N_{\rm ex}^2/\mathcal V).
$$
The second term cancels the direct normal term. Adding the <kinetic energy> yields the <number-conserving Bogoliubov quadratic Hamiltonian>
$$
\boxed{H_2=E_{\rm mf}+\sum_{\mathbf k\ne0}\left[A_k b^\dagger_{\mathbf k}b_{\mathbf k}+\frac{B_k}{2}\left(b^\dagger_{\mathbf k}b^\dagger_{-\mathbf k}+b_{-\mathbf k}b_{\mathbf k}\right)\right],\quad
E_{\rm mf}=\frac{\mathcal V V_0n^2}{2},\quad A_k=\epsilon_k+nV_k,\quad B_k=nV_k.}
$$
The constant is the condensate <mean-field theory> interaction <energy>; the normal term describes moving a particle out of the condensate, while the anomalous term creates or annihilates an opposite-<momentum> pair by exchanging two particles with the condensate. Total <particle number> remains fixed in the underlying theory; the condensate amplitudes in this notation are treated as classical amplitudes.
For <condensate fluctuation power counting>, put $\delta=N_{\rm ex}/N\ll1$. Formal interaction-amplitude counting gives $H_2-E_{\rm mf}\sim E_{\rm mf}\delta$, cubic terms $\sim E_{\rm mf}\delta^{3/2}$ and quartic terms $\sim E_{\rm mf}\delta^2$; these are not uniform bounds on individual modes. Thus the latter are down by $\sqrt\delta$ and $\delta$ relative to the quadratic interaction terms. The $N_0-1$ correction is relative $1/N$. Homogeneity eliminates terms linear in nonzero-<momentum> operators. Small depletion and weak interactions, not low <temperature> alone, justify the truncation. <Lecture Notes for Quantum Matter by Steven H. Simon>[§4.5, Bogoliubov expansion; fixed-number cancellation derived above]{ref}
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