Oh Boy! I had the same rash problems with both my girls. That was the first indication that they couldn't tolerate a new food. My girls would have open ulcers on their bottoms. I would try to wipe as carefully as possible, but you could see the layers of skin peel off. Very painful!
When my first daughter was born, they told me to put her on a soy formula, because I have sensitivies to milk. I did pump for the first 4 months(She couldn't latch on without gagging herself) After my milk slowed down she was on soy formula. She switched over to milk at a year, but only for a few months. She became allergic to milk at that point. She couldn't tolerate any milk proteins until she was about five years old. She wasn't allergic to lactose though. She could eat foods with lactose in the ingredients.
When I had my second daughter, I was told that they no longer recommend keeping a baby on soy as a preventitive, because they have a harder time with milk once they turn a year old and get switched over to milk. My second daughter was a horribly crabby baby. I also pumped with her, but discovered she wasn't tolerating my breast milk. I too had tried to modify my diet, to cut out the culprit. It wasn't until I had an infection and had to throw my milk out, and supplement formula inbetween milk from the freezer, that I discovered it was my milk she was allergic to. I started her strictly on milk formula at that time. She was a much happier baby. She too also developed an allergy to milk proteins as well after a year old. She couldn't digest milk products of any kind until she was about four years old. She too could have lactose.
What I did with my girls, was keep a notebook of all feedings and what time the symptoms showed up and how severe. This will come in handy if you need to discuss this with a doctor. I had gotten in to see a different doctor when our regular doctor was on vacation, and he told me that milk wouldn't be causing any of the symptoms in my daughter. (She was over a year old) He told me to go ahead and switch her to soy milk if it would make ME feel better. I did meet with our regular doctor when he got in town and I remembered to bring my notebook of symptoms along, and he said without a doubt, milk was the cause. And it was!
Try one kind of formula and see how your baby reacts. If the symptoms don't clear up, try the other.
Good luck!