Well you can bill the govt. but not sure you'll get paid ... ;-)
In general, and most people are surprised by this, you cannot sue the US govt in court. Well you can, but only in certain cases where it allows itself to be sued -- tort cases and some IP laws and even then you almost never get a trial by jury, only a federal judge.
So you'd have to construct this as a tort case and use Federal Tort Claims Act (FTCA) to your advantage. I suspect if you try that, FBI will invent some reason to arrest you or at least harrass you (interfering with an investigation, obstruction of justice, damage of federal property, etc etc...).
In general, and most people are surprised by this, you cannot sue the US govt in court. Well you can, but only in certain cases where it allows itself to be sued -- tort cases and some IP laws and even then you almost never get a trial by jury, only a federal judge.
So you'd have to construct this as a tort case and use Federal Tort Claims Act (FTCA) to your advantage. I suspect if you try that, FBI will invent some reason to arrest you or at least harrass you (interfering with an investigation, obstruction of justice, damage of federal property, etc etc...).