My DH is the Finance Manager for a hospital based unit that is 100% about mums and babies sleep and feeding (wishes he worked there 14 years ago) I was at a fundraising event the other day and so talking about babies and sleep etc.
BW talks a lot about baby temperament but in all honesty mothers temperament is huge... some Mums literally can't grasp the idea that routines and rituals need to change. Many are too tired to read books, particularly if reading has never been enjoyable... so it can become a cycle.
The nurse I was talking to said she always says 'short term it will be harder to make the changes than it is to keep going as you are, but long term Is what we are looking at"
I think with someone who is struggling but too tired to want anything but a magic solution, you have to show lots of empathy and Hint at options for when they are ready. Also your own experiences in a gentle way like, 'last night reminded me how much harder my child is when we don't do the routine properly.
Where DH works their methods have changed so much on the last couple of decades with more research, now they do a lot of tailoring to the Mum's personality, Mums who struggle with crying and aren't innately organised are often the ones with babies who don't sleeep well, as are th type A personalities that clock watch to the minute, and so they need to work differently with the Mums.
What is your friend and her baby's temperament (roughly)? As that might help with ideas of how you can be a middle person.