Here is my attempt to derive the residual reparameterization equation $$\partial_{\alpha}\xi_{\beta}+\partial_{\beta}\xi_{\alpha}=\Lambda \eta_{\alpha\beta}$$ For a reparameterization $x^{\alpha}\rightarrow x^{\alpha}+\epsilon\xi^{\alpha} \rightarrow(1)$. The pullback metric equation gives $$h_{\alpha\beta}(old)=\frac{\partial x^{\gamma}}{\partial x^{\alpha}}\frac{\partial x^{\delta}}{\partial x^{\beta}}h_{\gamma\delta}(new)$$ where $\gamma,\delta$ denotes new coordinate system. Using equation $(1)$ we get $$\eta_{\alpha\beta}=(\delta^{\gamma}_{\alpha}+\epsilon\partial_{\alpha}\xi^{\gamma})(\delta^{\delta}_{\beta}+\epsilon\partial_{\beta}\xi^{\delta})h_{\gamma\delta}$$ expanding the RHS upto $o(1)$ we get $$h_{\alpha\beta}+\epsilon\partial_{\beta}\xi^{\delta}h_{\alpha\delta}+\epsilon\partial_{\alpha}\xi^{\gamma}h_{\gamma\beta}$$ And this is where I got stuck since $h_{\alpha\beta}$ on RHS need not be a constant so I can't pull the $\delta$ index down had it been the case of covariant differentiation I could have done it but not with the partial derivatives. If somehow I can prove that $h_{\alpha\beta}$ can be expanded as $\eta_{\alpha\beta}+\epsilon (const)\eta_{\alpha\beta}$. Then I can prove the desired result but here I am assuming what I want to prove so I don't think this is the right direction to solve it.
I could do the pullback in opposite direction since these transformation are diffeomorphism. Then I would get $$h_{\alpha\beta}=\eta_{\alpha\beta}-\epsilon\partial_{\alpha}\xi_{\beta}-\epsilon\partial_{\beta}\xi_{\alpha}$$ But I don't know how to proceed from here either.
Also as a sidenote the way RHS appear reminds me of Lie derivative of metric tensor as a natural consequence can the technique of Lie derivative be useful in proving the required relation.
