Thank you all for the excellent suggestions!
@jagot, I realized I only need the time propagation, not the exponential explicitly. As such, your suggestion on the Crank–Nicolson method fits my particular use case perfectly, thanks!
@stevengj Somehow, I missed the decomposition you mentioned. This might be the best approach in the general case. In particular, the cis(::Hermitian) idea got me interested; I’ll make a PR for it soon.