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Write $\mathcal{V}=L^3$ and $N_0=N-N_{\mathrm{ex}}$, where $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. Macroscopic occupation permits $b_0,b_0^{†}\simeq \sqrt{N_0}$, with relative commutator corrections of order $1/N_0$. Keep the fixed-total-number correction before replacing $n_0$ by $n$.

The four-condensate term is $V_0{N}_0(N_0-1)/(2\mathcal{V})$. Its leading expansion gives $V_0{N}^2/(2\mathcal{V})-n{V}_0{N}_{\mathrm{ex}}$. Terms with two condensate operators give normal scattering $n(V_0+V_{\mathbf{k}})b_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$ and conversion of two condensate particles into opposite-momentum excitations, with coefficient $n{V}_{\mathbf{k}}/2$. The $n{V}_0$ normal contribution cancels against the change of condensate energy at fixed $N$. Hence the number-conserving Bogoliubov quadratic Hamiltonian isUse a real even interaction transform, $V_{\mathbf{k}}=V_{-\mathbf{k}}$, as for a real inversion-symmetric pair potential. The constant is the condensate interaction energy; the diagonal part combines particle kinetic energy with condensate scattering; the anomalous pair terms create and annihilate correlated opposite momenta while transferring particles to or from the condensate.

At fixed volume, counting replaced condensate operators gives quadratic vertices proportional to $N_0$, omitted cubic vertices proportional to $\sqrt{N_0}$ and quartic vertices with no condensate factor. There are also the finite-$N_0$ correction in $N_0(N_0-1)$ and corrections from the condensate-number expansion. Their physical control parameter is small depletion, $N_{\mathrm{ex}}/N\ll 1$, together with weak interaction; simply dropping them is not justified by low temperature in a strongly interacting gas. This is the weakly interacting Bose gas approximation.