i don't have a problem with your use of which or that, but the correction may have been offered for clarification. As it stands, it's not clear if the 'which' is going to refer to the labour pains, or back to her being a wonderful doula. Using 'that' would make it clear to me (as the reader) that (!) the next clause refers to the labour pains.
However, I don't know why I feel that way about it, and whether it's because of a rule hardwired into my brain or just instinct. Having never referred to Fowlers in my life, all I know is that as a reader I need it to be unambiguous. I can't say I've ever noticed an American distinction between the two that you describe and none of my American readers have ever commented on it.