I'm not trying to cover for him, I'm just trying to summarize what's in the book. Eugenics is bad. That he saw validity in it is bad. It is not heavily mentioned in the book, just some brief mentions. If you ctrl+f in the pdf, there are only 7 instances of the word outside of the endnotes. The most revelant parts being,

But Hirschfeld was also implicated in discriminatory practices, most obviously in relation to eugenics. Despite his later work on racism, published posthumously in 1938, he was in favor of the efforts of racial hygienists and eugenicists because like many scientists and political activists around 1900 he believed that these sciences could improve the health of the nation. (page 8)

According to Hirschfeld the [1889-1892 influenza] pandemic had “put all the cultured nations into the enormous grip of the East,” a turn of phrase that reveals his debts to contemporary debates about the impact and feared contamination of (German) civilization through encounters with people from the borders of Europe or beyond, debates that gained momentum during the colonial expansion of the German Empire. Hirschfeld’s doctoral thesis [about the symptoms of influenza] at first glance seems only tenuously linked to the German colonial project, but it was clearly framed in relation to the imperial and scientific discourses that gathered in its wake. The influence of these debates can be traced to Hirschfeld’s later work. He openly supported eugenics, for example, if not for “racial refinement,” then as a way of improving health via selective reproduction,* and returned to questions about the acclimatization of colonizers to the weather and (perceived and real) endemic diseases of the tropical regions as late as the 1930s, when he speculated about the suitability of the bodies of “the white man” and “the white woman” to life in the tropics. (pages 18-19)

* Endnote to the previous passage,

The phrase is “das Naturprinzip der Rassenveredlung” in the original. See Hirschfeld, Naturgesetze der Liebe, 132. There has been some debate about whether the support of eugenics by sexual reformers such as Hirschfeld directly contributed to the emergence of Nazism. Rather than such reductive and somewhat far-fetched arguments about a one-way flow of influence from homosexual culture to Nazism, it is more accurate to point out that both sexual reformers and right-wing hatemongers were animated by the scientific positivism of the turn of the nineteenth century. See, e.g., Marhoefer’s excellent critique of the debates in Sex and the Weimar Republic, 137.

Bauer moreso focuses on the fact that Hirschfeld's career was initially built by working with German soldiers and that he was exposed to the horrors of German colonialism, and the horrors of colonialism in general through the extremely racist world fairs of the time, yet he chose not to comment on them. Hirschfeld and the Institute were unique, first of their kind pioneers that started the work on developing a way to medically transition, along with all the advocacy for queer and reproductive rights that they did in tandem with the Wissenschaftlich-humanitäres Komitee, but they were very far from perfect. Hirschfeld was a cis white male doctor from Germany in the late 19th and early 20th century with all the expected baggage that would entail.