  The observability of Lyman-alpha emitting galaxies (LAEs) during the Epoch of
Reionization can provide a sensitive probe of the evolving neutral hydrogen gas
distribution, thus setting valuable constraints to distinguish different
reionization models. In this study, we utilize the new THESAN suite of
large-volume (95.5 cMpc) cosmological radiation-hydrodynamic simulations to
directly model the Ly$\alpha$ emission from individual galaxies and the
subsequent transmission through the intergalactic medium. THESAN combines the
AREPO-RT radiation-hydrodynamic solver with the IllustrisTNG galaxy formation
model and includes high- and medium-resolution simulations designed to
investigate the impacts of halo-mass-dependent escape fractions, alternative
dark matter models, and numerical convergence. We find important differences in
the Ly$\alpha$ transmission based on reionization history, bubble morphology,
frequency offset from line centre, and galaxy brightness. For a given global
neutral fraction, Ly$\alpha$ transmission reduces when low mass haloes dominate
reionization over high mass haloes. Furthermore, the variation across
sightlines for a single galaxy is greater than the variation across all
galaxies. This collectively affects the visibility of LAEs, directly impacting
observed Ly$\alpha$ luminosity functions (LFs). We employ Gaussian Process
Regression using SWIFTEmulator to rapidly constrain an empirical model for dust
escape fractions and emergent spectral line profiles to match observed LFs. We
find that dust strongly impacts the Ly$\alpha$ transmission and covering
fractions of $M_{UV} < -19$ galaxies in $M_{vir} > 10^{11} {\rm M}_{\odot}$
haloes, such that the dominant mode of removing Ly$\alpha$ photons in non-LAEs
changes from low IGM transmission to high dust absorption around $z \sim 7$.
