My reply based on two years of baby travel but not internationally nor with special dietary needs: You're at the mercy of whoever is screening you. Bring the soy milk, but be prepared to toss it. Soy milk isn't exactly prescription medication. I can see someone giving you a hard time despite the fact your kids need it and despite the doctor's note. Expect inconsistency, as the homeland security secretary says.
Alternatives:
a) Powdered soy milk. (Isn't "real" soy milk from a solid anyway?) We carry powdered cow's milk with us. It comes in convenient paper packets. Take an empty quart water bottle through security with you, fill it at the water fountain and you're in business.
b) Does ANA cater soy milk aboard? It's an Asian airline. Worth an inquiry.
c) Skip the milk altogether and stick to juice and water. Problem solved. Maybe even let the older one have her first taste of S-O-D-A? It's gotta happen sometime.
d) Break the 8.25-oz containers into smaller 3.2-oz containers. (Big hassle.)
e) Pour the milk into breast milk containers and say it's breast milk. (Lying to federal officers not recommended.)
f) Recon IAD before your trip to see if perhaps soy milk is sold airside (no doubt for $2/fl oz.)
On the return out of SIN, I'd be shocked if you couldn't buy soy milk airside there. Again, not cheap.