That's unfortunate about her reaction, it does sound like she can't tolerate the milk quite yet if she's reacting like this with yogurt and butter. I know what you mean about the birthday cake, I really wanted my dd to have a taste of her birthday cake the day of her party, I ended up giving her a very tiny piece to try (it was day 3 after her milk related runny bm), I probably shouldn't have but I did. You're right, it's not a bad thing that she can't have cake, it's us moms that want them to taste cake, not them, they don't know what they're missing and there's lots of time when they're older to eat cake!
I would try to stay away from soy sauce (if it's the normal soy sauce) as it's very high in salt, I would instead start with small amounts of soy milk when testing her on soy, was there anything else in the burgers that she could have reacted to or do you think for sure it was the soy?