I ended up sitting in the waiting room of the doctor's surgery for 2 hours before my appointment for my son, as he was almost exactly as you described. I went early and sat there so that they heard what he was like, and how nothing calmed him.
My doctor, after hearing how distressed he and I both were, referred me to a paediatric allergy specialist, and even rang them herself to make an emergency appointment.
My son was diagnosed with allergies to dairy AND soy. As it was all dairy (so cow, goat, etc) and soy, all the alternative formulas (the lactose free, soy etc) that were tried didn't work, as he was still allergic to them.
The allergy specialist prescribed a new formula based on rice, it is expensive and requires a prescription here (and it actually tastes quite nice, all the other older ones taste feral). But it worked!
My son is now like a different child, and is sleeping 10 - 12 hours a night (a big change from 20 minute cat naps a few times a day or night), and the crying and pain have stopped.
I was a first time mum, and got told often that "all babies cry, it's normal", I did the waiting room camp out because no one would listen or believe it was more than normal crying every baby does.
You know something is wrong, because you are the one with your child! You also know it is different from the other children you have. Please keep pushing for a doctor to listen - it sounds like there is definitely something wrong with your poor Bub, and both of you need some help.
Hugs from me for both your Bub and yourself.