Non-linear omitted term in Poisson vs OLS regression

Question

I am considering two models.

One is a Poisson model where the true relationship is:

$$E( y \mid x,z)=exp(bx+cx \times z + dz+ f(x))$$

The other is a linear model:

$$E( y \mid x,z)=bx+cx \times z + dz + f(x)$$

In both, x and z are independent exogeneous variables and $f(x)$ is some deterministic non-linear function of x (specifically, in my case, $f(x)=\log \Phi(x)$ with $\Phi(x)$ being the standard normal CDF).

I believe that omitting the f(x) term biases the estimate of the $b$ term in both the linear and Poisson models. (This inconsistency is suggested for the Poisson case by an answer here.) Simulations, however, indicate that the estimate of the $c$ and $d$ terms are consistent in the OLS model, but not in the Poisson model.

I am wondering what could explain the difference. It is difficult to me to understand why these terms would be biased in the Poisson case given that $x \times z$ and $f(x)$ terms seem independent conditional on $x$. Intuitively, this tells me that once $x$ is included in the Poisson estimation, $ x \times z$ should not be biased. (Same goes for the main effect of $z$.) This is likely a mistaken intuition however, possibly related to differences in how the Frisch-Waugh-Lovell theorem applies in the two cases?