Successful potty training is not strictly age-dependent; kids train somewhere between 18 months and 5 years, depending on an array of physical, nervous, and emotional factors, and cultural/familial patterns. Every child is different. Some kids need more time to get the pooping thing worked through, and will withhold if they feel pressured, resulting in constipation, painful elimination, further withholding, and possible encopresis (which is a serious medical problem that's hard to correct). These difficulties in turn slow down the poop training, so pressuring a child before she's willing or able may end up lengthening the whole process, and certainly adding stress for both child and parents. (Your loss of patience, and your daughter's pain, are HUGE warning signs.)
I've known a number of moms who do let their kids use a diaper for pooping for however long they need it. When these children are given the time they require to sort out the sensations and their feelings about the ongoing responsibility to get to the potty on time every time, they do eventually decide they can handle it and develop a genuine willingness. At that point, training is often complete, and child-led, in a day or a week.
Because your daughter has had the added stress of painful withholding, it may take her awhile to do the necessary sorting out. That's really not her fault, and if you keep urging her to do it your way, the whole situation is likely to become worse.
It's great to have a child who's fully trained, and with another baby coming, I can sure understand your eagerness. But this is something that the child works through, not the parent. The best scenario for potty-training is what some researchers call potty "learning," which is completely child-led and parent-supported. When approached from this natural, unpressured direction, regression when a new baby arrives is far less common.
The latest I personally have known a child (a boy) to fully day train was age 4 + 10 months, and that child is still not night trained at 7 years. But two different pediatricians convinced his parents to follow his lead, and assured them that when his body and nervous system have matured enough, he'll be night trained, too.