One can certainly appeal it all the way to the full board.

Prior to that, it's appealed to an ALJ.

After the full board, it's appealed to the US Court of Appeals.

In the interim, and forever after, it's part of one's FAA record, and before one gets a change to appeal, one is the subject of an FAA Order; eg, enforcement action. Conviction first, then the opportunity to defend one's self.

Any defense up to that point is simply an evidence-gathering exercise to be used against the airman. If it doesn't go to enforcement action, the FAA places the letter of investigation in the airman's file, and that will do nearly as much damage as enforcement action. Even if the FAA generates a letter following the initial investigation to state that evidence has not been found to further the enforcement process, the FAA will state it in terms that establish guilt. We have been unable to find evidence that you violated x regulation, but doing x is a violation of that regulation and that letter will remain in the airman's file. It's damaging, without question, and the FAA knows this.