Kate coming back to work is a theme in this novel. In the first book in this series, Kate's partner (Lee, not Al) got shot in the back. Kate had never officially announced her domestic situation at work. Because the crime involved a very famous person, and a very notorious earlier crime, Kate's sexuality and domestic arrangement suddenly became very, very public. Kate had gotten "outed" in the most spectacular way possible.
What seems to bother Kate most is not that her sexuality was made public, but that her personal life in general was made public. She resents it when Brother Erasmus tells her he's sorry that Lee got hurt.
Getting "outed" isn't just having someone else reveal your sexual orientation. You can get "outed" any number of ways. Jews got "outed" in the 1930s and 1940s. Liberals get "outed" as Republicans. Republicans get "outed" as liberals. Any time someone reveals personal information about you, that others don't really have any right to know, you've been outed.
Often it's just embarrassing. Bed wetters get "outed".
Sometimes it's much more serious. "Outing" someone in the military who is homosexual can ruin their career.
About ten years ago, David Brin wrote an amazing book titled The Transparent Society, where he suggested that the best way to solve the problems of uncomfortable information is just to make everything public. It would entirely eliminate the issue of blackmail. If there's something you're embarrassed about, either don't do it, or make it public. If there's something you want kept to yourself, then don't talk about it. If you don't advertise it around, it won't be an issue.
If everyone knew everyone else's "dirty little secrets", there'd be a lot less embarrassment. "Yeah, I wet the bed occasionally. So does Bob Superstar, Mary Rockstar, and George Politician. What of it?"
Having an affair with a staff person? Either get permission from everyone involved, or just DON'T DO IT.
What do you think is better? Keeping the secret, or "tell the truth and shame the devil?"