= Solution
At order $\lambda_0g_0$, use one anisotropic quartic vertex and one $O(N)$-invariant quartic vertex joined by two fast propagators. With four equal external component labels, the bubble renormalizes $\lambda$; with external labels in two equal pairs, it also renormalizes $g$. The anisotropic vertex fixes the component index carried by the internal lines, so none of these mixed diagrams contains an independent component loop. Consequently no correction proportional to $N$ occurs at order $\lambda_0^2$ or $\lambda_0g_0$.
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