Hmmm; check with the Post Office again. I read a while ago that they had a flat rate for APOs but that may have been the case only during the holidays. Last fall I got free packaging supplies and they sent them to me at no charge. The number for that (the "Military Pack") was 1-800-610-8734. Maybe there is current information about rates there, too.
"The only downside was that a 20 pound package to an eastern APO address from Nevada ran $20. Too bad the P.O. doesn't have a "Humanitarian Rate". "
The flat rate boxes are $7.70. The boxes aren't very big either.