Reddit reviews Mothers on Trial: The Battle for Children and Custody

For the Prohibition aspect, you can take a look a many of the biographies of early feminist icons, but Susan B. Anthony and Carrie Nation are perfect examples. The ties between women's suffragism and women's temperance movements were legion. The first part of Ken Burns' Prohibition documentary has a decent overview, and this account from a modern feminist has a few more details.

For the "Tender Years" doctrine, the first chapter of Mothers on Trial by Phyllis Chesler has a readable and brief feminist version of the history leading to modern custody law.

For the last, the link between early feminism and eugenics is explored in a good research paper by Mary Zeigler available online. There was also a dust-up over some university with a building named for Carrie Nation a few years back, over the some blatant white supremacist bits in her letters, which I'm sure is Google-able.