I'm not sure in what context you have been seeing $\beta \sim g^{-2}$, but to me (lattice-QCD person) it seems like you're getting two different $\beta$'s confused. In Lattice-QCD, which is naturally formulated in Euclidean spacetime, the continuum action is discretised onto a lattice by introducing $SU(3)$-valued wilson lines on each of the edges of the lattice. The action is written as:
$$S[U] = \frac{\beta}{3} \sum_{n \in \Lambda} \sum_{\mu < \nu} \mathrm{Re}\mathrm{Tr}(1 - U_{\mu \nu}(n)) = \frac{a^4}{2 g^2} \sum_{n \in \Lambda} \sum_{\mu,\nu} \mathrm{Tr}(F_{\mu \nu}(n)^2) + O(a^2) $$
where $\Lambda$ indicates the set of lattice points, $\mu,\nu$ are euclidean spacetime directions, and $U_{\mu\nu}(n)$ is the plaquette in the $\mu,\nu$-th direction, $\beta$ is defined to be $6/g^2$, and $a$ is the lattice spacing (see (2.54,3.4) of Gattringer and Lang). The path integral is then expressed as:
$$Z[U] = \int dU e^{-S[U]}$$
where $dU$ is the product haar measure over all the links. Because it is formulated in Euclidean spacetime, this looks just like a statistical mechanics system, where $\beta \to \infty$ is the "$T \to 0$ limit" (where the path integral limits to sampling over only configurations with very small actions), this is also the continuum limit. There are two different perturbative expansions available, the strong-coupling expansions (large $g$, small $\beta$) by expanding observables in powers of $\beta$, or weak-coupling expansion (also known as lattice perturbation theory, by expanding in small $g$, or large $\beta$).
Note however that this $\beta$ has nothing to do with the actual temperature of the Euclidean path integral. By placing the lattice in a box with finite extent in the Euclidean time direction (and periodic boundary conditions), then you are actually simulating at a temperature $T$ corresponding to $1/L$, where $L$ is the length of the box in the Euclidean time direction. Sometimes people refer to this length $L$ by the variable $\beta$, but this is not the same $\beta$ as the one that appears in the lattice action.
