Germany doesn't require any vaccines, so the miltiary cannot make you get any shots at all. If you are worried about measles in Europe you can just go to any German doctor and get the measles only shot for your children and the military will never know. It costs very little since it's socialized medicine so if you have no insurance there are standard rates that apply (ca. 50 Euros for visit and then the shot runs at about 10-40 EUros, depending on which one you get).
The military can give you a religious exemption to all vaccines, but you cannot do some and not others. It's all or nothing. Past shots don't count, you might have turned religious recently, kwim? So I would claim a religious exemption and then privately do shots in Europe that I want without the military or any US registry ever knowing.