We investigate in this work families $(u_\epsilon)_{\epsilon>0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type: $$\Delta_g u_\epsilon+ h_\epsilon u_\epsilon= |u_{\epsilon}|^{p_\epsilon-2} u_\epsilon $$ in a closed manifold $(M,g)$, where $h_\epsilon$ converges to $h$ in $C^1(M)$ and $p_\epsilon$ converges to $2^\star:=2n/(n-2)$. Assuming that $(u_\epsilon)_{\epsilon >0}$ blows-up as \emph{a single sign-changing bubble}, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between $h$, the geometry of $(M,g)$ and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of $u_\epsilon$.