The single forecast solution in a nonlinear dynamic model will not be unbiased. There's really no way to get an single unbiased forecast by inverting the nonlinear expression in the way that we do.
What people tend to do in this case is to perform stochastic simulation of the dynamic model and then take moments. Potter has a brief discussion in the context of more complicated time series modelshttps://www.newyorkfed.org/medialibrary ... s/sr87.pdf
There are other references in that paper too.
have a bit of a discussion too under section 4.2 -- Monte Carlo forecasts
In EViews, the model object has built in stochastic simulation in which we'll generate a bunch of forecasts after adding disturbances and then collect moments. It's pretty straightforward and a search of the documentation or the forum should get you going.