I like your idea of getting the shower stool nice and soapy before your mother sits down. Make sure it is not too slippery. The nurse who taught me always
stressed the importance of occasional soaking for cleaning privates. You could use a perineal cleanser which has aloe vera. I have a bottle of Aloe Vesta which is soothing to the skin, dissolves fecal matter and does not have to be rinsed off. It does tend to be a little expensive. But there are different brands.
Another way is to let your mother sit on the toilet with a walker in front. Have baby wipes, warm wash clothes, zinc cream or some moisture barrier as well as the aloe vera cleanser on hand. When your mother finishes with her bowel movement, have her stand and hold onto the walker. Wipe her with a baby wipe and then wash her well with the cleanser and warm wash cloth. You can give her the cloth and ask her to wash her private parts. If she can't, she may let you. After she is as clean as you can get her, rub her bottom with the moisture barrier.
The cleanser and warm water will be soothing for her and probably will make you feel better as well.
This method can also be accomplished while she is lying on the bed. You could have a big towel or bath blanket spread out on the bed. After her shower, have her lie down on her side. You can cover her with a big towel.
Start by washing her back side. Then you could move around to the other side of the bed and tell her that you will be cleaning her privates. "Is that okay, mom?" You can ask. If she says no, then ask her if she would like to do it herself.
With these methods in mind you might want to go to a medical supply store, or a big drug store that has a medical supply section. See what products are available.
Also, you might be able to get a nurse from the visiting nurse service or a licensed home health care agency to come to your home and train you on good way to wash your mother. (Medicare should pay for this family training.) There is nothing like the comfort of a kind nurse with good skills.
Your mother is very lucky to have such a wonderful daughter who is paying attention to the details of caring with such kindness.