Those of us who have been stationed overseas are aware of one caveat that most others are not. While it is true that a military base is U S territory and, in theory, it is no different than shipping to any of the states here at home there is one additional wrinkle. According to the Status of Forces Agreement, nothing can be shipped into a military base which is illegal in the host country. The theory is that a serviceman could just walk it off base and sell it in the black market.
So the underlying question is whether you could legally ship whatever it is to the whatever the host country is. If not, then you can not ship to an APO in that country. I do believe that there are some restrictions on electronic devices. That requires separate research. But an APO address in and of itself will not absolve you of responsibility.
As for the battery - from what I have read if the battery is actually installed in the device you are OK to ship. But if the battery is loose, even if it is included in the package, then you can not.
"Laissez-faire capitalism (AKA The Great Material Continuum) is the only social system based on the recognition of individual rights and, therefore, the only system that bans force from social relationships." ~ Ayn Rand
