You're getting wrapped around the axle on things that are of no consequence.
To answer your question, they CAN sponsor an H1B. They may even be cap-exempt. Feds have always been. The local guys, have gotten similar treatment in recent year.
The bad news for you is nobody HAS to sponsor you. In fact, the whole point of the H1B program is stacked against them doing so unless your skills are so unique that they wouldn't be able to find anybody else to do the job. That's almost never the case with government agencies. If their hiring, someone usually wants the job.
Unless you're some sort of rare scientist or some professional ( like a doctor) who they are in short supply of in the public sector, you don't have a chance. It's a far cry from your F-1/OPT stuff.