I definitely wouldn't survive this if I were still a fed. Back in my day (says the boomer) you could at least just take the mandatory class, sit there and keep your mouth shut and not participate, and as long as they got you on the sign-in sheet, you were okay. Or take the online course that just required you read it, without you having to input any answers.
It sounds like they're heading for mandatory active participation now. I'd probably be written up in the first five minutes for refusing to give my pronouns.