In the 2010s, Kashaev, Luo, and Vartanov introduced a Turaev--Viro--type partition function associated with any triangulation of a cusped 3-manifold admitting angle structures. Analogous to the relationship between Turaev--Viro and Reshetikhin--Turaev invariants, the KLV partition function is expected to be closely related to the partition function in Teichm\"uller TQFT. In this talk, I will describe how, for geometric triangulations, the asymptotics of the partition function capture both the hyperbolic volume and the Hodgson--Kricker--Siejakowski $\tau$-invariant. The main technique involves generalizing the Jones function in Teichm\"uller TQFT previously constructed by Fathi Ben Aribi and myself. As a corollary, we establish a relationship between the KLV partition function and the partition function in Teichm\"uller TQFT for knot complements in $S^3$.