Scattering processes featuring the strong interactions can be studied using lattice QCD by means of the Lüscher formalism. This approach relies on analyticity and unitarity of the $S$-matrix to relate infinite-volume scattering amplitudes to finite-volume energy levels. However, lattice QCD simulations employing rooted staggered fermions manifest unitarity violation as an $\mathcal{O}(a^2)$ lattice artifact. Moreover, the meson sector of this theory contains multiple non-mass-degenerate pions (due to the so-called taste splitting), which only reduce to the physical pion in the continuum limit. These features restrict the applicability of the Lüscher formalism to staggered lattice data at non-zero lattice spacing. Hence, in this work, we discuss two complementary approaches to deal with the challenges of extracting $\pi\pi$ scattering amplitudes from lattice QCD with staggered quarks: (1) using the corresponding effective theory, Rooted Staggered Chiral Perturbation Theory, to calculate one-loop amplitudes for the first time. These amplitudes can be used to explicitly check the validity of the quantization condition. And (2) generalizing the formalism to incorporate taste-splitting as well as fourth-rooting effects. We focus on the simpler case of $\pi\pi$ scattering in the isospin-2 channel, and discuss prospects for other channels.