“it has a different effect because it is pig’s milk”
There is no pig’s milk or horse’s milk sold in the United States. Most dairy farms do not have any pigs or horses at all! It is illegal to put any milk from any animal other than a cow, sheep, or goat into any product that is transported across state lines for sale in the United States, and if anyone ever tried to do that they would get caught and it would cause a major scandal. In fact, despite draconian inspections that are far more frequent and strict than the supervision provided by the major kashrut agencies to food processors that don’t sell meat, there has never been such a scandal. There ARE a few farms that sell camel’s milk, but usually you have to buy it on the farm and it is clearly labeled.
Milk DOES require rabbinic supervision, as all milk sold in the US contains nutritional supplements (at least Vitamin D). And it needs to be pasteurized unless you enjoy eating bacteria culture media. About 15 years ago, one dairy in the Northeast decided to “go organic” and added a nutritional supplement from shark tissue. It is the only time I have ever seen milk that was actually non-kosher in the US. Their major competitor attacked their loss of kashrut certification in television and radio ads!
If you are getting sick from milk, consider that you might be lactose intolerant. The Lactaid brand and similar knockoffs might work for you. (There is even Lactaid ice cream.) They are kosher chalav stam.
Now, if anyone can explain why we are so worried about chalav Yisrael when it is at most a d’rabbanan, but ignore yashan even though it is a d’oratia….