I HIGHLY recommend not getting into your reasoning with the doctor.
"it is against our religious beliefs" is really the only thing to say. Depending on the state you live in, and especially with the military involved, that is the ONLY way to avoid harassment. State "it is against my religious beliefs" and if pressured say "my religious beliefs are not up for discussion, thank you".
One of the potential problems for getting into ANY type of scientific or medical reasoning, is that your exemption could actually be jeopardized. In states which have ONLY a religious exemption, if you start getting into medical/scientific/philosophical reason why you don't want to vax, then your religious exemption could be turned down on the grounds that your reasons for not vaxxing aren't truly religious. This is only the case in certain states, New York and New Jersey being 2 of the most notable, but regardless, there is no downside to using religious beliefs. It is the one thing you cant be harassed about, the one thing all states but 2 must accept. PLUS...the doc can't argue woth you on your religious beliefs. If you start blabbering about toxicity, aluminum, formeldahyde, reactions, the body's natural immune system, etc..you are giving him/her something to argue with....and, unless you are a REALLY educated person in the areas of immunology/virology/biology, etc, and VERY knowledgeable about what you are saying, chances are the doc will be able to win hands down in a verbal sparring session about vaccines. Don't argue. "against my religious beliefs"
In public at least, the only thing you should say is that they are against your religious beliefs. To the docs, to the military people, to the school officials. Against my religious beliefs.