I would to like confirm that,
non-linear moving average method( Lan/Meyer-Gohde) evaluate the policy function at the ergodic mean or stochastic steady state while usuall perturbation evaluate the policy function around the deterministic steady state using state space method. And Lan/Meyer-Gohde do not use state space rather they develop Euler equation error method.

Their NLMA add in(link given below) for dynare do evaluation of policy function around ergodic mean not the stochastic steady state

My understanding is that this is not the case. A true approximation at a point that is not the deterministic steady state would lead to effects of uncertainty already at first order. But they still need to perform a third order approximation to capture the effect of uncertainty shocks.