What are good, reliable databases of atomic spectra?

I am sure that this won't cover everything you need, but a "competitor" to the NIST database that is used a lot in astrophysics is the Vienna Atomic Line Database VALD database of information on atomic and molecular transitions.

As I say, I think you will only get a subset of what you need here because it is focused on the data required to predict the emergent spectrum from a stellar atmosphere.

There are several mirror sites in different countries!