The classical limit of a QFT in deformation quantization

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I'm not sure if I understand the following correctly, but it seems to have appeared implicitly in my studies.

If $Z_{cl}$, $Z_p$, and $Z_\hbar$ are the classical, perturbative to arbitrary order, and full nonperturbative solutions to the generating function of some quantum field theory, then the three functions arise as (approximate) solutions to a Schrödinger-like equation of the form
$$y'(x)=f(x)\star y(x)$$
for some definition of the star product. To my knowledge $Z_p$ is reproduced (derived?) by the solution to the above ODE with Picard's method, and moreover if one takes the limit $\hbar\to 0$ either beforehand or in the perturbative solution, one obtains $Z_{cl}$. However if one solves it first then takes the limit, there is some ambiguity with convention and the classical solution is not obtained:
$$\lim_{\hbar\to 0}Z_\hbar \neq Z_{cl} = \lim_{\hbar\to 0 } Z_p$$
I've heard the first equation as an explanation for why the perturbation series is riddled with divergences (i.e. the generating function has a discontinuity at $\hbar=0$, among other reasons), but haven't seen something substantive on the topic.

How does one recover the classical solution and perturbative series given $Z_\hbar$? A Taylor series doesn't seem to work because of the discontinuity.

This pathology can be rewritten in terms of the star product itself i.e. star product doesn't nicely converge $\implies$ $Z$ doesn't converge nicely either (at least for one definition of the star product).
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