What is the state in the WRT TQFT associated to a handlebody?

(Making this an answer, since it will be a little long, even though it doesn't actually answer the question.)

One thing to be aware of is that there are multiple ways to talk about this same TQFT. In particular, Reshetikhin and Touraev took a much more combinatorial point of view in defining the 3-manifold invariants. There's an entirely different way of getting at the vector space associated to a surface $\Sigma$ from the combinatorial point of view. You can take a handlebody $H$ with boundary $\Sigma$, pick a trivalent graph spine $\Gamma$ for $H$, and look at the vector space spanned by admissible colorings on $\Gamma$. You then have to work a little bit to see that different choices of $H$ and $\Gamma$ give canonically isomorphic vector spaces.

This is quite different from the description in terms of line bundles, as you described. It's not obvious that the two descriptions are isomorphic, and I'm not sure where to point you to get a proof that they are isomorphic. (Of course Witten gives a beautiful physics argument for isomorphism.)

I can think of two cases where the Witten-Reshetikhin-Turaev vector $Z_k(Y^3)\in Z_k(\Sigma)$ has been connected to a Lagrangian state as you described:

-Laurent Charles and Julien Marché showed that the WRT vector of the figure eight knot complement $Z_k(E_K)$ is a Lagrangian state concentrating on $\mathcal{M}(E_K)=\mathrm{Hom}(\pi_1 E_K, \mathrm{SU}_2)/\mathrm{SU}_2$ and used this to prove Witten's asymptotic expansion conjecture for Dehn-fillings of the figure eight. (see here part1 and part2 )

-In my thesis, I considered the vectors $Z_k(H,\Gamma)$ where $H$ is a handlebody and $\Gamma$ is a colored trivalent banded graph in $H$ that is a spine of $H.$ Such vectors form a basis of $Z_k(\Sigma),$ and the vector $Z_k(H)$ of the empty handlebody is obtained taking all colors=0. I showed for generic colors, the vectors $Z_k(H,\Gamma)$ are Lagrangian states concentrating on subsets of the form $\lbrace \mathrm{Tr}\rho(C_i)=x_i \rbrace$ where the curves $C_i$ bound disks in $H$ that are dual to the edges of $\Gamma$.

Both results were obtained from the combinatorial definition of TQFT (Reshetikhin-Turaev, or rather Blanchet-Habegger-Masbaum-Vogel), after choosing an isomorphism $Z_k(\Sigma) \rightarrow H^1(\mathcal{M}(\Sigma),\mathcal{L}^k)$ that turns curve operators in TQFT into Toeplitz operators whose principal symbols are trace functions on the moduli space $\mathcal{M}(\Sigma)$.

Although the problem of showing that the WRT vector $Z_k(H)$ of a handlebody concentrates on the character variety of $H$ seem reasonable, one gets into trouble because of the singularities of $\mathcal{M}(\Sigma)=\mathrm{Hom}(\pi_1\Sigma,\mathrm{SU}_2)/\mathrm{SU}_2,$ and also somewhat because one has to work at the critical levels of the principal symbols of the curve operators.

Laurent and Julien's work avoided this difficulty because $\mathcal{M}(T^2)$ is not really singular, and in my paper because I only considered the geometric quantization of dense open sets of $\mathcal{M}(\Sigma)$ and not the whole moduli space.

It is my feeling that if one was able to give a clean answer to your question, one would not be too far from proving Witten's asymptotic expansion conjecture, at least for $3$-manifolds with a "generic" Heegaard splitting.