There are cool examples already for finite group: Dijkgraaf and Witten recover and generalize a combinatorial formula due to Burnside using the TFT formalism.

It's probably worth elaborating on this. Let $\Sigma_g$ denote a compact orientable surface of genus $g$ and let $G$ denote a finite group. The problem is to compute $|\text{Hom}(\pi_1(\Sigma_g), G)|$; for example, when $g = 1$ this is the number of pairs of commuting elements of $G$, which is $|G|$ times the number of conjugacy classes of $G$, or equivalently $|G|$ times the number of irreducible representations. The formula, which I know as Mednykh's formula, gives the answer as

$$\frac{|\text{Hom}(\pi_1(\Sigma_g), G)|}{|G|} = \sum_V \left( \frac{\dim V}{|G|} \right)^{\chi(\Sigma_g)}.$$

where the sum ranges over all irreducible representations $V$ of $G$. Note that when $g = 1$ we recover the above result and when $g = 0$ we recover the fact that $|G| = \sum_V (\dim V)^2$. This result was used here to show that $\pi_1(\Sigma_g)$ is not a free group ($g \ge 1$).

There's some leeway in terms of how you juggle the powers of $|G|$ here, but I prefer this statement because the LHS is the groupoid cardinality of the "moduli groupoid" of principal $G$-bundles on $\Sigma_g$; this is the number that directly appears in the TFT story, as the value of "untwisted" Dijkgraaf-Witten theory on $\Sigma_g$.

The TFT proof can be found, for example, here, although that document commits the terrible sin of having no diagrams. The important diagram to draw here involves breaking up a surface of genus $g$ as a composite, in the cobordism category, of a cup, $g$ copies of the "tube of genus $1$," and a cap, and the important computation involves applying a TFT to this composite.

The TFT proof generalizes directly to the case of "twisted" Dijkgraaf-Witten theory, where a class $\alpha \in H^2(G, \mathbb{C}^{\times})$ appears. The LHS is now a sum over homomorphisms $\pi_1(\Sigma_g) \to G$ weighted in a manner determined by $\alpha$ while the RHS is now a sum over the projective irreducible representations corresponding to $\alpha$. I don't know if this more general result has a character-theoretic proof.