Yes, these are rational functions, with the poles you suggested.
One step back. The Exponential Generating Function of the polynomials $\big\{\sum_{k\ge0} b(n,k)t^k \big\}_{n\ge0}$ is $$\sum_{n\ge0}\Big(\sum_{k\ge0} b(n,k) t^k \Big)\frac{x^n}{n!} =e^{t(e^x-1-x)}$$ by the combinatorial meaning of the exponential of an EGF. Hence also, for any $k\ge0$, $$ \sum_{n\ge0} b(n,k) \frac{x^n}{n!}=\frac{(e^x-1-x)^k}{k!}.$$
To convert the latter into an ordinary GF we may apply a Laplace transform
$$\sum_{n\ge0} b(n,k) x^n =\frac{1}{k!x} \int_0^{\infty} (e^s-1-s)^k e^{-\frac{s}{x}}ds ,$$ which clearly produces a rational function: if we expand the term $(e^s-1-s)^k$ and integrate with a linear change of variables we get a finite sum $$\sum_{n\ge0} b(n,k) x^n =\frac{(-1)^k}{x}\sum_{i\ge0, j\ge0\atop i+j\le k} \frac{(-1)^j}{i!j!}\Big(\frac{x}{1-jx}\Big)^{k-i-j+1} ,$$ which is a rational function of the form you suggested in last edit.
Rmk: here the Laplace transform can be applied formally, as all needed identities holds in a formal context; also, for $|x|<1/k$ the convergence of the integrals and series are ensured.