That seems like the job for the NSA or CIA though. To flag these individuals.

There is an FBI database they use to do background checks. Last time I heard there is a huge backlog and it’s delaying cases by a lot.

It’s actually Department of State databases, specifically, CLASS (Consular Lookout and Support System) and the TIDE (Terrorist Identities Datamart Environment) systems that are used. There is also (as you mentioned,) an FBI fingerprint and name check that is completed, but that’s for domestic (US) criminal records and not specific to overseas terrorism.