Organic merely tells you that the wheat that was grown, was grown in a more environmentally friendly way, ie. without the aid of pesticides. That on its own will have no effect on noodle stability.
They may be poor quality noodles, have gluten level issues, are they also wholewheat, do your regular udon have some form of strengthening additive like with ramen? Many noodles have kansui added, or whatever passes for that chemically (sodium carbonate) added to strengthen the dough. Of course it may not have been kneaded enough, too old, too dry.
Soy milk is bitter. Enzymes in the beans (lipoxygenase) combine with fats in the presence of water to produce what is usually described as a "beany flavor"; bitter and grassy.
The solution to this problem, although not done in many traditional soy milk preparations, is to cook the soy milk long enough to destroy the enzyme. Many, but not all, soy milk manufacturers will cook the milk as well as add a lot of flavorings to mask the bean flavor.
When enough lemon juice is added to soy milk it will coagulate but in my experience it does not produce flavors more bitter than the milk itself. I have, however, found that, regardless of coagulant, soy milk declines in quality very quickly and, even when I thoroughly boil it to deactivate enzymes and "beany" compounds, it will go from sweet to bitter in just a few days. I have also had homemade soy milk start to turn after a relatively short amount of time developing rancid flavors or activity that implied fermentation such as independently souring or clabbering.
I don't have any proof but based on my own experience I suspect that the lemon is not the culprit. If your milk has become so bitter as to be inedible then I think your soy milk may have spoiled.
As for the foaming- in your comment below you mentioned that the soymilk in question was Silk. Silk is one of the most modified soymilks- they definitely err on the side of flavor rather than simplicity and have a lot of additives for flavor and texture.
One of the additives is Calcium Carbonate. Calcium Carbonate gently coagulates soymilk and is sometimes used for making silken tofu. In Silk it is no doubt used to make the product more creamy.
The existence of Calcium Carbonate in your milk could explain your foaming as it can form CO2 in the presence of acids.
Notes
Soy milk does not coagulate well with lemon juice. It requires a lot of acid to coagulate and so is overly sour and it abruptly produces a very fragile curd. I agree that the advice that you read must have been for baking. If you can't use real buttermilk then in such a dish I would recommend silken tofu for the creaminess and don't worry about the acidity.
In general the way to temper bitterness is to add salt.
As a rule of thumb food that is bitter is often poisonous. I don't imply that that is the case here but you should trust your tongue better and not feel forced to eat questionable food.
Best Answer
I'm at a bit of a loss providing advice on this issue because I've never heard of it (short of burning the ingredients in the dish). There's a very, very slight bitterness in proper ramen-style noodles because of the alkali content of kansui (potassium carbonate and sodium carbonate or bicarbonate in water solution), but in typical noodles of reasonable quality I would expect the bitterness to be lower than, say, the outer skin of a pretzel.
To me it sounds like you've stumbled on poor quality noodles, or you're hypersensitive to the very mild alkali content. You might consider looking for Chinese egg noodles instead, which have a similar color but may have less or no kansui, or you might try a completely different noodle, such as udon, and make yaki-udon instead. Alternatively, consider trying a different brand of noodles, or a different form (dry instead of refrigerated, frozen instead of dry, etc.) One local brand of refrigerated noodles meant for yakisoba available where I live (Seattle) has been so inconsistent in quality for me that I skip it and go straight to the (probably imported) frozen ones in our market. (Dry ones will require brief boiling), but in my case our complaint was about texture, not bitterness. Switching out for a different supplier may resolve your issue.
ETA: It occurs to me that if you were starting from dry noodles, you may encounter rancidity with unusually old ones. That's a bit more aggressive than just bitterness, but I've seen it on occasion with long-forgotten packaged noodles; it was obvious before cooking, though, so I've just tossed them.