If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-Viro construction for two Morita equivalent fusion categories gives the same 3d TQFT. Equivalently said, if $\mathcal{C}$ has Froboenius algebra objects $A$ we can contruct Morita equivalent categories by condensing $A$, which turns out the be the category of bimodules of $A$, and this corresponds to all possible Lagrangian algebra objects of the Drinfeld center.
My question is whether this extends to fusion 2-cateogries and 4d TQFTs. I know that there are state-sum constructions of 4d TQFTs from the datum of a braided fusion 1-category (which can be understood as a fusion 2-category with one simple object), like the Crane-Yetter state-sum, or more generally from the datum of a spherical fusion 2-category (by Douglas and Reutter). It seems that still the braided fusion 2-category associated to these 4d TQFT is the Drinfeld center of the fusion 2-category we started with (correct me if I am wrong), and this is an invariant under Morita equivalences (for instance https://arxiv.org/abs/2211.04917). However in the case of fusion 1-cateogries the Drinfeld center is, in a certain sense, the unique invariant of the Morita equivalence classes, since for any Froboenius algebras of $\mathcal{C}$ there is a Lagrangian algebra object of $Z(\mathcal{C})$. It is not clear to me whether this is true also in the case of fusion 2-cateogries. Actually it seems to me that the Drinfeld center of a fusion 2-cateogory is not the only invariants of its Morita equivalence class.
One of the origins of my confusion is the following. The Crane-Yetter state sum for a modular tensor category produces a 4d TQFT which is an invertible theory, and can be thought as the anomaly inflow theory for the 3d Reshetikhin-Turaev theory of the starting MTC. However if the MTC has some commutative Froboenius algebra objects we could condense them producing Morita equivalent categories, and I would expect the Drinfeld center to have a corresponding Lagrangian algebra object. But this cannot be the case if the 4d TQFT is invertible... In other words all the MTCs have the same Drinfeld center (it is trivial) but not all of them are Morita equivalent. Is there something beyond the Drinfeld center which differentiate the various Morita equivalence classes?
(Summary of the comments:) The input of Turaev-Viro model is a spherical fusion 1-category $\mathcal{C}$, and it produces a fully extended 3d TQFT in the Morita 3-category of tensor categories c.f. DSPS whose value on the circle $S^1$ gives you the Drinfeld center $\mathcal{Z}(\mathcal{C})$. In comparison, Reshetikhin-Turaev model takes a modular tensor category $\mathcal{M}$ as input, and produces a 3d TQFT extended down to 1d, whose value on the circle $S^1$ gives you the original MTC $\mathcal{M}$.
The former 3d TQFT can be viewed as anomaly-free, while the later 3d TQFT is anomalous, whose anomaly can be described by a 4d TQFT, which is exactly the Crane-Yetter model you mentioned at the beginning. This 4d TQFT is (probably not surprising) anomaly-free again, which means you can extend it down to a point. Hypothetically this 4d TQFT should be realized as a symmetric monoidal 4-functor from the cobordism 4-category to the 4-category $\mathbf{4Vect}$.
The target 4-category might have many inequivalant models, for example one could consider all linear 3-categories. Meanwhile, Cobordism Hypothesis tells us that nevertheless their full subcategory of fully dualizable objects are always the same.
Going back to the Crane-Yetter model, when your input is a MTC $\mathcal{M}$, the value assigned to the point by the corresponding 4d TQFT is the linear 3-category $\Sigma^2 \mathcal{M} := \mathbf{Mod}(\mathbf{Mod}(\mathcal{M}))$, or equivalently, the Morita class of monoidal 2-category $\mathbf{Mod}(\mathcal{M})$ of module categories over $\mathcal{M}$, with monoidal product given by the relative Deligne tensor product $\boxtimes_{\mathcal{M}}$.
When people say that Crane-Yetter model gives you an invertible 4d TQFT, they mean the value it assigned to the point, $\Sigma^2 \mathcal{M}$, is an invertible object under tensor product of linear 3-categories. In this paper of Brochier, Jordan, Safronov, Snyder BJSS, the authors proved that this group of invertible linear 3-categories is the so-called Witt group of non-degenerate braided fusion categories defined by Davydov, Müger, Nikshych and Ostrik DMNO. If we ask for the value assigned by 4d Crane-Yetter TQFT to the circle $S^1$, then it is the trivial linear 2-category $\mathbf{2Vect}$; mathematically, one has $\mathscr{Z}(\mathbf{Mod}(\mathcal{M})) = \mathbf{2Vect}$ as braided fusion 2-categories.
Question 1. Does Drinfeld center of a fusion 2-category determine its Morita class?
The answer to this question is yes-and-no. It is still fairly obvious that Drinfeld center $\mathscr{Z}(\mathfrak{C})$ is a Morita invariant for fusion 2-category $\mathfrak{C}$, as one can realize it as $\mathbf{End}_{\mathfrak{C} \boxtimes \mathfrak{C}^{1mp}}(\mathfrak{C})$. However, as we have observed from Crane-Yetter TQFT story, one could have fusion 2-category $\mathbf{Mod}(\mathcal{M})$ not Morita equivalent to $\mathbf{2Vect}$, as long as MTC $\mathcal{M}$ has non-trivial Witt class. In the language of condensed matter physics, you may have two anomaly-free 3d topological orders with no gapped domain wall between them; in comparison, there always exists a gapped domain wall between two anomaly-free 2d topological orders.
Question 2. What is the relationship between algebras in a fusion 2-category $\mathfrak{C}$ and Lagrangian algebras in the Drinfeld center $\mathscr{Z}(\mathfrak{C})$?
In the setting of tensor 1-categories, their relationship is established by the construction of full center (Davydov et al.). Of course one could try to ask for a categorification in this setting of fusion 2-categories. The desired theory of full center should tell you that
If you take $\mathfrak{C} = \mathbf{2Vect}$, then this reproduces the classical statement (due to Etingof et al.) that Morita equivalences between fusion categories are in one-to-one correspondence with braided equivalences between their Drinfeld centers. On the other hand, take $\mathfrak{C} = \mathbf{2Vect}_G$ for some finite group $G$, then this claims the correspondence between Morita equivalences between $G$-graded fusion categories and their full centers in $\mathscr{Z}(\mathbf{2Vect}_G)$, which are $G$-crossed braided fusion categories.
Finally, maybe it is beneficial to remark on the classification of fusion 2-categories in general. Firstly, Décoppet has proven that any fusion 2-category is Morita equivalent to a strongly fusion 2-category, i.e. a fusion 2-category equivalent to either of the two cases:
Thence, for any fusion 2-category $\mathfrak{C}$, we can always find a braided equivalence $\mathscr{Z}(\mathfrak{C}) \simeq \mathscr{Z}(\mathbf{2Vect}_G^\pi)$ (in the bosonic case) or $\mathscr{Z}(\mathfrak{C}) \simeq \mathscr{Z}(\mathbf{2sVect}_G^\varpi)$ (in the fermionic case). Using results from DX, this corresponds to find a Lagrangian algebra $L$ in $\mathscr{Z}(\mathbf{2Vect}_G^\pi)$ or $\mathscr{Z}(\mathbf{2sVect}_G^\varpi)$. Thus the classification of fusion 2-categories reduced to