And they can do pretty much anything, now that they're exempt from search, seizure or any serious investigation.

Yes, but "they" are just a bureau that can only "do pretty much anything" if they can persuade local law enforcement to behave that way. They don't have any James Bonds of their own and local law enforcement is not covered by the immunity.

As for the archives: any locally-derived information is already available (from source) under the Freedom of Information Act. The main effect of immunity is on information shared by counterparts abroad, information to which US citizens do not have any legal right and for which creation of extra rights would not be in US interests, because foreign countries would be foolish to share sensitive information if it can be picked apart and exposed under US law.

I'd also observe that FBI Director Robert Mueller III told a senate committee last year that the FBI needs "global reach" to fight cyber-crime and terrorism and that co-operation with "law enforcement partners" gives it "the means to leverage the collective resources of many countries". He's talking about standardized surveillance technologies that the US wants to be able to access. Perhaps a few local changes/concessions may be justified if it helps persuade other countries to agree?