I just read the 5th amendment with the wikipedia link you provided and I cannot find where it said that it does not protect the president or the military.
Perjury is a felony, at least here in Hawaii. Not sure that is something you wanna just think of as minor.
"No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury,
except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb;
nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."
Now that I read it yet again though, maybe the military exception clause only applies to the first part of the Fifth Amendment, which isn't the self-incriminating part we are concerned with here.
Perjury is a serious charge but I think it should be considered what was the original activity that was lied about. If Bill Clinton said under oath "I drank orange juice this morning" but then a videotape surfaced of him drinking apple juice, why should he be impeached? That's how I view what happened to him...am I still missing something? The Fifth Amendment, Executive Privilege, Article 31 of Military Justice, etc should only provide him further protections.