Brandon was cognitively aware (which for Biden was not very high on his best days) when he committed the crimes. Because he knew what he was doing at the time of the crimes, he should be prosecuted. His current mental state can be a factor in the sentencing. It is the job of the prosecutor and the judge to instruct the jury that the defendant's current mental state is not a factor in determining his guilt or innocence for crimes committed in the past but is a factor for sentencing.
Prosecutor Hur should just say "fine, if you want to insist that you are mentally competent, then I'll go to a grand jury and get an indictment." (Or whatever the next step would be to move the process forward.)