You Need Sideways Access
The US intel community (and most intel communities for other major powers) are big in theory, but widely distributed into small groups. The FBI, for instance, only has about 60,000 people in it. What's more, each individual office has it's own territory, and crossing from one Area of Responsibility into another is a faux-pa at best, and a serious policy violation at worst. The individual offices are small enough (from a few dozen to a thousand-odd, and that includes "non-agent" staff) that everyone knows everyone. So your fake FBI employee would have to pretend to be from another office. BUT the further you get from the area's actual office the more suspicious it becomes, while conversely the closer you are the more likely you are going to blow it via not knowing who you should know. (ie: somebody claiming to be from the New York Office in the LA office's territory is going to raise a trillion red flags. Someone claiming to be from San Diego's office will raise a couple dozen red flags, and might get burned because agents from LA will know many agents from San Diego.)
Instead, I suggest two things:
1: Get your agency Universal Database Access. The USG has hundreds of databases with bespoke access. This one only the FBI has access too, this one DIA uses, this one CIA and TSA use but FBI has no access. 89
This one FBI and DEA have access to but CIA doesn't, this one only state and local law enforcement have etc etc etc. However "read only" access would be trivially simple to set up, and would resolve 90% of your potential problems. After all, you don't need access to a CIA office if you can already see the recordings the CIA made and documented in their file! (this obviously wouldn't work as well in the 80s. But that just gives you some hand-wavy "back in my day" talk from older MIB Agents about having to fake being DEA Atlanta to send a file request to DEA New York and then intercept it etc etc. But nowadays everything is digital which means your life is easy!)
2: If you are out and about and run into, say, a DEA agent, you don't show them a DEA badge. You show them a FBI badge, or an ICE badge, or whatever. Which depends on circumstance. Some agencies have excellent "no questions asked" relationships with other agencies, either nationally (Say maybe DEA and FBI work well together all over, but the DEA HATES some other agency and only helps them grudgingly through formal channels) or at an office level (Maybe the FBI and CIA don't get along, but the New York FBI office gets along well with the CIA because of "reasons". Or maybe the CIA and FBI work great together EXCEPT the NY FBI office.) If you train YOUR agents in these sorts of byzantine quasi-political trust systems, you can easily get tons of cooperation without risking mis-steps based on not knowing a person's name you should or not knowing a normal bit of agency trivia.
As a sidenote, small obscure organizations are also great to fake. The Small Business Administration has a dozen-odd "Special Agents" that technically have all the investigative powers as FBI agents, but specific to SBA-related crimes. Faking one of these guys with a humble attitude could easily get you "in" with more "prestigious" organizations, as even a high-ranking FBI person is unlikely to know anything about his supposed equals in the SBA! 89