Can someone who isn't a member of the military server the role of prosecutor in a case heard by a military tribunal?
Eeeeh... not really but not for the reasons you thinks. Military Tribunals are distinct from a Court Martial in that they are tribunals where the Judge is a military officer in an inquisitorial hearing (as opposed to an advesarial hearing, which is the standard use in American Court rooms). In the inquisitorial hearing, the judge serves the role of both Judge (trier of Law) and Jury (trier of fact). It's also distinct from a bench trial in that while the judge in a bench trial is also the jury, bench trials are still adversarial, which means the judge does not ask questions to the witnesses independent of the prosecutor and defense. In Inquisitorial systems they do, so the judge may, ask any witness questions independent of either advocate's line of questioning.
Military Tribunals are very rare in the United States and made rare following the SCOTUS decision Ex Parte Millgan which held that military tribunals to civillians are unconstitution when courts are still functioning. As such, any modern military tribunal conducted by the United States must be done to an enemy combatant only unless there is no functioning court in the jurisdiction. Essentailly, while this doesn't ban a civilian prosecutor, it does severely limit the situations where such a prosecutor can enter the tribunal (to say nothing about the fact that military tribunals are so beyond the norm of the U.S. Jurisprudence, they are generally unfavored by the public and frowned upon by the chain of command).
As stated early, the United States Military has it's own independent court system called the Branch's Judge Advocate General Corps (JAG Corps or JAG... yes, the one depecited in the tv show under that name. No, not a British Car manufactor). JAGs refer to both the judges (usually a Colonel or Naval Captain rank or higher) and the attorneys (both defense and prosecution) collectively. They are especially trained to handle law among servicemen and women, but will often handle legal services for servicemen/women's civilian dependent family members as well. It's preferable to try enemy combatants in a JAG court hearing whenever possible and occasionally they may try civillians as well (in cases where the federal government has exclusive jurisdiction AND the crime occured on a U.S. Military Base. Often this is done if a civvilian broke a law in a host country that they can't be extradited for (usually the base is overseas and the U.S. does not think the civvilian family member will recieve a fair trial for a crime if they hand them over to local legal systems. Again, this is very rare because usually families don't live on these bases for this very reason.). In any situation, there is no case where a JAG hearing is prosecuted by a civillian prosecutor (Though a civilian defense attorney can argue the defendant's case if the Judge allows it. Sometimes they were JAGs but have since left the military service (you can't be a JAG unless you are a barred attorney in one state and only then will you be allowed to study the additional courses needed to become a JAG).