Washing your mother's hair in the shower sounds like a good idea. Her weight, however, could be to much for you ( and your back) to handle. It is possible to wash your mother's hair while she's in bed.
Why not contact a local hair dresser and see if she might be willing to come to your home and shampoo and style your mother's hair?
Or, if your mother has a wheelchair, you might be able to position her head over the sink, and then wash her hair yourself, without having to get her into the shower.