The other classical limit of a quantum enveloping algebra?

Let $mathbb K$ be a field (of characteristic 0, say), $mathfrak g$ a Lie bialgebra over $mathbb K$, and $mathcal U mathfrak g$ its usual universal enveloping algebra. Then the coalgebra structure on $mathfrak g$ is equivalent to a co-Poisson structure on $mathcal U mathfrak g$, i.e. a map $hatdelta : mathcal U mathfrak g to (mathcal U mathfrak g)^{otimes 2}$ satisfying some axioms. A formal quantization of $g$ is a Hopf algebra $mathcal U_hbar mathfrak g$ over $mathbb K[[hbar]]$ (topologically free as a $mathbb K[[hbar]]$-module) that deforms $mathcal U mathfrak g$, in the sense that it comes with an isomorphism $mathcal U_hbar mathfrak g / hbar mathcal U_hbar mathfrak g cong mathcal U mathfrak g$, and moreover that deforms the comultiplication in the direction of $hatdelta$: $$Delta = Delta_0 + hbar hatdelta + O(hbar^2),$$ where $Delta$ is the comultiplication on $mathcal U_hbar mathfrak g$ and $Delta_0$ is the (trivial, i.e. which $mathfrak g$ is primitive) comultiplication on $mathcal Umathfrak g$. This makes precise the "classical limit" criterion: "$lim_{hbar to 0} mathcal U_hbar mathfrak g = mathcal U mathfrak g$"

I am wondering about "the other" classical limit of $mathcal U_hbar mathfrak g$. Recall that $mathcal Umathfrak g$ is filtered by declaring that $mathbb K hookrightarrow mathcal Umathfrak g$ has degree $0$ and that $mathfrak g hookrightarrow mathcal Umathfrak g$ has degree $leq 1$ (this generates $mathcal Umathfrak g$, and so defines the filtration on everything). Then the associated graded algebra of $mathcal Umathfrak g$ is the symmetric (i.e. polynomial) algebra $mathcal Smathfrak g$. On the other hand, the Lie structure on $mathfrak g$ induces a Poisson structure on $mathcal Smathfrak g$, one should understand $mathcal U mathfrak g$ as a "quantization" of $mathcal Smathfrak g$ in the direction of the Poisson structure. Alternately, let $k$ range over non-zero elements of $mathbb K$, and consider the endomorphism of $mathfrak g$ given by multiplication by $k$. Then for $x,y in mathfrak g$, we have $[kx,ky] = k(k[x,y])$. Let $mathfrak g_k$ be $mathfrak g$ with $[,]_k = k[,]$. Then $lim_{kto 0} mathcal Umathfrak g_k = mathcal Smathfrak g$ with the desired Poisson structure.

I know that there are functorial quantizations of Lie bialgebras, and these quantizations give rise to the Drinfeld-Jimbo quantum groups. So presumably I can just stick $mathfrak g_k$ into one of these, and watch what happens, but these functors are hard to compute with, in the sense that I don't know any of them explicitly. So:

How should I understand the "other" classical limit of $mathcal U_hbar mathfrak g$, the one that gives a commutative (but not cocommutative) algebra?

If there is any order to the world, in the finite-dimensional case it should give the dual to $mathcal U(mathfrak g^*)$, where $mathfrak g^*$ is the Lie algebra with bracket given by the Lie cobracket on $mathfrak g$. Indeed, B. Enriquez has a series of papers (which I'm in the process of reading) with abstracts like "functorial quantization that respects duals and doubles".

On answer that does not work: there is no non-trivial filtered $hbar$-formal deformation of $mathcal Umathfrak g$. If you demand that the comultiplication $Delta$ respect the filtration on $mathcal Umathfrak g otimes mathbb K[[hbar]]$ and that $Delta = Delta_0 + O(hbar)$, then the coassociativity constraints imply that $Delta = Delta_0$.

This makes it hard to do the $mathfrak g mapsto mathfrak g_k$ trick, as well. The most naive thing gives terms of degree $k^{-1}$ in the description of the comultiplication.