when I was alittle kid and had a Betsy Wetsy doll I put her in a pan and washed her with Ivory soap which is what was used on me at the time. You have to be careful with the eyes if they are the opening/closing kind. Just put the doll in a sitting position the way you would a baby and don't get soap in her eyes.
I like ZUD cleanser. It comes in a yellow plastic bottle. It's a white pourable liquid, but thick. Like a cream almost. It's good for scrubbing. I like it for bathtubs, sinks and showers. Gets rust out too.