I find that a lot of dysphoria involves what I see in the mirror: the more hormones feminized my face, the less dysphoric I felt looking at myself. So I think that self-assessment of gender plays a large role in gender dysphoria.
I bring this up because very, very few animal species can recognize their own faces in a mirror. It's not even a universal trait among great apes. This mirror test is a way of measuring self-awareness. It may be that for gender identity to develop, an animal needs a high degree of self-awareness, and very few species have brains that meet that minimum requirement.
If I were looking for gender dysphoria in another species, I would look chimpanzees and bonobos. They probably have the necessary level of self-awareness to generate a gender identity. How that gender identity would present itself in the wild, however, is not clear. That's a question for a primatologist.