(Neighbourhood of $S$) Given $0<\epsilon\ll 1$ and $-3+\epsilon<\delta<0$, let $f$ be an $S^{1}$-invariant function compactly supported in $M^{-}_{\nu}\cap\{R\lesssim A^{-1/2}\}$ with bound $\left\lVert f\right\rVert_{C^{\alpha}_{\delta}}\leq 1$. Then there is a function $u$ on $\mathcal{B}$ supported in $\{R\lesssim A^{1/4}\}$, with decay estimates $\left\lVert u\right\rVert_{C^{2,\alpha}_{-1+\epsilon}}\leq CA^{-1},$ such that $\left\lVert\Delta_{g^{(2)}}u-f\right\rVert_{C^{\alpha}_{\delta}}\ll 1.$