As far as I know, there's no specific "DSM test." When I first went to my gender clinic, they did have me take a battery of psych tests, and the testing psychologist spent about two hours going over them with me. Most of them didn't relate to gender ID -- they were nearly all personality tests. My therapist and I talked very briefly about the results and then moved on to the important stuff, like... my feelings. But that's just one clinic, albeit one that does follow the WPATH standards of care pretty closely.
The WPATH standards state, pretty much, that any mental illness should be under control before someone starts to transition, but it doesn't list any that would rule out transitioning. I think that the testing my clinic did related to two things: it was a way of streamlining the "getting to know me" part of the process, and because it's part of an academic system, some of the tests were part of whatever research various faculty members were doing.
I don't recall the WPATH SoC saying that psych testing is necessary for a diagnosis of gender dysphoria.