There are 2 separate issues here: the diet and the lying.
1) If you're visiting someone, you eat what they serve. It's not a restaurant that you can order food. You don't get to be picky. Same goes when they visit you, they don't get to demand a meat meal. Your dietary choices are your problem, don't make it other people's problem.
2) The other issue is lying. You asked, they agreed, they lied. There are no excuses for that even without making you sick. The fact that it made you sick makes point #1 questionable. We don't really know enough to judge them. Maybe they agreed happily, planning to lie from the beginning. Maybe you were grinding them for months and they've finally buckled in and lied to get you off their backs. But the point is that it doesn't matter for the end result.
I don't wish to eat there anymore like this, but they are direct family and obviously my partner does not want a fight with his parents. His solution was to just bring Tupperware with our own food or only to invite them to our place and have us cook.
That's actually how you solve this kind of things. If your lifestyles are incompatible, it's better to dial down contact rather than engage in a cold war. It's not a fight, it's a fact - you just can't eat there. Make shorter visits that doesn't involve eating or go out together to a place where both parties can order something they're comfortable with. Weaning takes many forms and levels, this is one of them.
I feel like we should be able to discuss this as adults to adults. I don't feel that lying about these kinds of things is a very adult decision. What would be the best approach to discuss this without it escalating?
Well, from my point #1 it's not an adult thing to come to someone's home and have demands. Even if that's parent's home. It's not an adult thing to lie about what was actually served either. I don't really think there is a room for discussion. However you present it, either as you stopping pressuring them to cater to your needs or you holding them accountable for what they did, the end result is same: you don't eat what they cook. It's not end of the world.
The actual compromise would be both sides stepping back a bit: you accepting some meat, they accepting no milk. But I don't think it's possible here. Elimination diet is pretty much defeated by weekly breaches and veganism is perceived as a kind of cult by non-vegans. You sound like a moderate person, but it doesn't stop the other side from perceiving you through the "weird vegan" stereotype. I'm only talking about veganism, even though you carefully avoided the word, because the stereotypes about vegans still affect you, even if you're not identifying with the movement. Parent's insistence on not changing their cooking once a week may stem from them seeing it as defending their lifestyle against alien culture.
Or they might be just picky eaters. Like my father-in-law who eats pretty much 3-4 things and wouldn't even try anything else. For this reason alone, my mother-in-law is very rigid when it comes to menu. They're more stubborn than kids and they have the means to have it their way. It took my wife many years to learn that it's not even worth trying.
For a person who's making their signature dish in a same way for decades, asking to substitute an ingredient usually is a grave offense. They'll fight defending the original recipe. The soy milk could be that.
There is also another angle of you "driving" your partner away from his mother. He used to like her cooking, now "you've made him" reject it. That's subconscious insult to every mother.
For all reasons, I suggest trying the neutral ground of a restaurant or takeout food for a while. It can make everyone cool down and open up.