This is a problem a lot of parents face, and the best treatment is most definitely not convenient. I'm sure you're already doing a lot of this, but...
The short answer: do everything you can to minimize all sources of irritation from mechanical (wipe/wash cloth/diaper/cotton balls) to chemical (urine/soaps/alcohol/chemicals/stool), decrease moisture (air dry), and protect.
Diaper rash or irritant diaper dermatitis is a nonspecific medical term describing inflammatory skin reactions, something you know your daughter has (if her skin is red, there's inflammation.) Hers is likely a chemical burn, but that doesn't mean other things aren't also affecting her skin's ability to heal itself.
- Urine hurts her skin, too. So as long as she's red, no wet diaper (even the absorbant kind.) Better - and faster for healing - is no diaper at all.
Exposure of urine to feces causes the formation of ammonia via degradation by the ureases of fecal microbes. The resulting pH increase can reactivate fecal proteases and lipases, and thus lead to attack of the
corneocyte proteins and interstitial lipid lamellae of the stratum corneum, which results in an impaired barrier function. Hydrated skin is also more prone to mechanical damage and chafing of the skin as it shows an increased coefficient of friction and may allow irritants to penetrate the stratum corneum more easily. These observations contributed to the understanding that good control of skin wetness and pH are critical for maintaining skin health in the diaper area.
(That just says urine + any stool is much worse than either one alone.)
Every time she wets or soils herself, wash her bottom with a mild (pH around 4-7) cleanser in a bath of lukewarm to coolish (as tolerated) water (instead of a wipe, which is abrasive - even if only microabrasive, it causes this already damaged skin further damage.) Let the water, the mild surfactant and your hand to the work of cleaning her skin. (While it seems unnecessary to wash her bottom after urinating every time, see above. This is a different scenario than everyday care.)
Gently pat dry (you can use a blow dryer set to low and hold it far from her bottom) and let her stay bare bottomed for a while. Drying the skin discourages microorganisms associated with diaper rash, minimizes the effect chemicals can have on the skin, and allows more repair to take place.
After she's had a chance to thoroughly dry out, apply some A&D ointment (not sticky like zinc ointment and easier to remove, but still protective) or zinc oxide ointment.
Leave her bare bottomed. That way, you'll know the moment she soils, then repeat the above.
If you really must use something to decrease the redness - and it sounds like it would be helpful given her pain after passing any stool - 1% hydrocortisone cream 4 x/day (for one day only) is recommended. (That really should be sufficient. If you still feel she needs extra help, use twice the second day. Longer than that should be by her doctor's permission.) Apply it to her dried skin, allow to absorb/dry, then apply A&D ointment.
Symptomatic care (when she's howling) can be a cool compress with a soft cloth and some soothing lotion.
You should see an improvement in a day, and more complete resolution in 2 days, enough to go back to her usual routine.
"Ideally, first-line therapy for diaper dermatitis is zinc oxide ointment. The safest over-the-counter (OTC) emollient available for newborns is pure white petrolatum ointment. Another safe alternative is Aquaphor ointment, which is principally composed of white petrolatum, mineral oil, and lanolin. It is more expensive than pure white petrolatum ointment. Zinc oxide has antiseptic and astringent properties. It plays a significant role in wound healing with low risk for allergic or contact dermatitis. To remove zinc oxide from the skin, mineral oil is more effective and easier than soap and water." - From reference below
Diaper Dermatitis Treatment & Management