We have $0<\inf_{g\in \mathcal{T}} L_g(\gamma) \leq \sup_{g\in \mathcal{T}} L_g(\gamma) <\infty$. In fact, there should be a universal bound on the ratio $\sup_{g\in \mathcal{T}} L_g(\gamma)/ \inf_{g\in \mathcal{T}} L_g(\gamma)$ for all $\gamma \in \pi_1 M$.
This follows from a theorem of Belegradek, who proves that the class of such metrics (actually, with just a fixed fundamental group $\pi$) is precompact in the Lipschitz topology. In particular, all such metrics are uniformly bi-Lipschitz, and thus one has the bound on the ratio between maximal and minimal lengths.