I don't quite know how this answer fits with André's, and there are certainly a bunch of subtleties I'm unaware of, but:

In the spirit of the cobordism hypothesis, to get a 3d TFT you need to attach a monoidal category $C$ to the point. In order to go up to dimension 3, this category should, as you say, satisfies some dualizability conditions which amongs to say that $C$ is actually fusion. This is basically the result of Douglas-Schommer-Pries-Snyder you mention, and what you get is the Turaev-Viro TFT associated with $C$

Then what is attached to the circle has a canonical structure of a modular category, and is nothing but the Drinfeld center $Z(C)$ of $C$.

Roughly, you can get Reshetikhin--Turaev TFT by starting directly from the circle, replacing $Z(C)$ by any modular category, and then reconstructing the higher dimensions "in the same way". This explain why Turaev-Viro of $C$ and Reshetikhin-Turaev of $Z(C)$ essentially coincides.

Therefore, if your modular category is not the center of some fusion category, I think you can't go down to the point.

The category of $U_q$-modules you mention is certainly not the center of any other category. In fact there is a conjecture that these are the building blocks of all modular categories which are not equivalent to the center of a fusion category. Its relation with the category of $LG$-modules at level $k$ has been made precise by Kazdhan--Lusztig and Finkelberg.

All of this is somehow related to the fact that the RT TFT usually have a so-called anomaly and so is strictly speaking not quite an actual 3d TFT. The "right" way to recast this in the framework of the cobordism hypothesis is to see the RT construction as some sort of boundary condition of an extended, honest 4d TFT. See e.g. What's the right way to think about "anomalies" in 3d TQFTs? or Freed-Teleman on relative quantum field theory.

I don't quite know how this answer fits with André's, and there are certainly a bunch of subtleties I'm unaware of, but:

In the spirit of the cobordism hypothesis, to get a 3d TFT you need to attach a monoidal category $C$ to the point. In order to go up to dimension 3, this category should, as you say, satisfies some dualizability conditions which amongs to say that $C$ is actually fusion. This is basically the result of Douglas-Schommer-Pries-Snyder you mention, and what you get is the Turaev-Viro TFT associated with $C$

Then what is attached to the circle has a canonical structure of a modular category, and is nothing but the Drinfeld center $Z(C)$ of $C$.

Roughly, you can get Reshetikhin--Turaev TFT by starting directly from the circle, replacing $Z(C)$ by any modular category, and then reconstructing the higher dimensions "in the same way". This explain why Turaev-Viro of $C$ and Reshetikhin-Turaev of $Z(C)$ essentially coincides.

Therefore, if your modular category is not the center of some fusion category, I think you can't go down to the point.

The category of $U_q$-modules you mention is certainly not the center of any other category. In fact there is a conjecture that these are the building blocks of all modular categories which are not equivalent to the center of a fusion category. Its relation with the category of $LG$-modules at level $k$ has been made precise by Kazdhan--Lusztig and Finkelberg.

All of this is somehow related to the fact that the RT TFT usually have a so-called anomaly and so is strictly speaking not quite an actual 3d TFT. The "right" way to recast this in the framework of the cobordism hypothesis is to see the RT construction as some sort of boundary condition of an extended, honest 4d TFT. See e.g. What's the right way to think about "anomalies" in 3d TQFTs? or Freed-Teleman on relative quantum field theory.