How do you know she's not lactose intolerant? Did you make that.judgement based.on.her reaction to soy? (That she still has diarrhea.) Getting a rash when starting milk may just be a coincidence.
How old is she? If she's younger than one or just a couple of months older, her digestive system may not be mature enough for cows' milk. Same for soy. Other foods can cause diarrhea as well.
If she is eating solids her body may be reacting to the food. She may have one or more food intolerances or allergies. She may be allergic to the diapers or the wipes. We can become allergic to things that caused no problems before.
Are you sure it's not a common diaper rash? How often do you change her diaper? What treatment do you use? Another possibility is a yeast infection. What does the rash look like. Yeast infections spread over the whole bottom. The ones I've seen have blisters that break and cause pain.
There are several alternatives to cow's milk if milk turns out to be the cause. Check with the doctor before trying nut milks. She may be too young for them. My granddaughter thrived on rice milk. The allergist suggested using natural fruit drinks some of the time. She still prefers fruit juice over.milk, though she outgrew the milk protein allergy around 6.
Does your daughter have a medical diagnosis that excluded milk intolerance? She may actually be allergic to milk protein. If she hasn't seen the doctor for a diagnosis, I recommend doing that. There is no simple answer.
First you need to know what is causing the diaper rash? Babies do have diarrhea that causes a rash and then they don't. Unless the diarrhea lasts long term, I wouldn't consider that it was caused by milk or soy or an intolerance to food. I would limit the number of foods and drinks and keep a food diary. Perhaps only feed two things for a few days or use the BRAT diet.
A tummy bug can cause diarrhea. If that's the cause, yogurt with live cultures can help. Probiotics are good for her even if they don't help with the rash.
You may never know what caused the diarrhea. Short term diarrhea, by itself, is not cause for alarm. How long has she had diarrhea? How much of her bottom does it cover? How red/angry does it look?
I would try a BRAT diet. Banana, rice, applesauce, and toast. Those foods help reduce diarrhea. TheSE foods, banana especially, can cause constipation for some babies. Be sure to give her enough water.
My 5 month old granddaughter has diarrhea at times. She does not like it and let's us know right away. Frequent diaper changes keep the rash mostly away. Leaving the bottom open to air helps.