Well, the FBI should decline any request for an investigation. Is Kavanaugh alleged to have committed a federal crime? No! Then the FBI should say no! They've also already vetted him up to 6 times. If the Dems wanted an FBI more thorough background investigation, they should have requested it months ago, and the answer should be No! again. Why should Kavanaugh be subjected to another background investigation when the D's obviously committed mis or non feasance with respect to that.
That assumes that the investigation would only repeat work already done, which I've read only went back to 18. If they want to go further back, that would be additional work, but is non-standard work and would have to be requested as such.
I'd compare it to a standard "X point inspection" being the background checks, but now they're asking that the fuel system be specifically investigated. Or maybe check the bumper to make sure that nobody stashed bricks of cocaine in it. The X point inspection should catch any potential problems, but this would be like seeing some potential problem that the inspection didn't catch, so look closer.
I'll point out that I don't think that having the professionals take a closer look is a bad thing.
That said, the accusations against Kavanaugh add up to a whole lot of nothing. We have a lack of collaboration, the best another "witness" has is that he attended a party where rapes might of occurred, but it reads like the whole Rolling Stones accusation.