One simple way of producing symmetric monoidal $(\infty,n)$-categories with all duals is to form $n$-fold spans/correspondences, hence an (∞,n)-category of spans $Span_n(\mathbf{H})$ in some ambient $\infty$-topos $\mathbf{H}$.

This is discussed around section 3.2 in Jacob Lurie's "On the classification of TFTs".

In fact in $Span_n(\mathbf{H})$ every object is fully self-dual even. For low $n$ this is spelled out a bit at the beginning of these notes here

For $X \in \mathbf{H} \hookrightarrow Span_n(\mathbf{H})$ any object, the corresponding invariant assigned to a closed framed $n$-manifold $\Sigma$ is $X^{\Pi(\Sigma)}$, where $\Pi(\Sigma) \in \infty Grpd \simeq L_{whe} sSet$ is the homotopy type of $\Sigma$ and the exponential notation denotes the powering of $\mathbf{H}$ over $\infty Grpd$.

While these are not the *quantum* invariants that you are looking for, this are in some precise sense the PREquantum invariants of a local field with moduli sstack $X$, before quantization. An exposition of this is in the lecture notes *geometry of physics* in the section on *prequantum field theory*

A slight variant of this (also discussed there in more detail) works as follows: for $G \in Grp(\mathbf{H})$ an abelian $\infty$-group object, also the $(\infty,n)$-category $Span_n(\mathbf{H}_{/G})$ of $n$-fold spans in the slice $\infty$-topos over $G$ is symmetric monoidal with all duals. Objects are now maps $\exp(i S) : X \to G$ and their duals are now

$$
\exp(-i S) : X \to G
$$

(using the inversion operation in $G$). As the notation suggests, the manifold invariant induced by that now are prequantum fields equipped with a local action functional.

These are *still* not the interesting quantum invariant that you are looking for, but this is now that data which upon "quantization" should give rise to them.

For discrete higher gauge theories (Dijkgraaf-Witten-type theories) this is indicated in sections 3 and 8 of Freed-Hopkins-Lurie-Teleman.