Beyond my bailiwick as to why they would prescribe gender transition to cure gender dysphoria. I'm not into that portion of the literature. However, it is clear there are people who benefit from the procedure, and it is clear that there are people who don't and are noticeably worse off.
The relevant question is how many people are benefited by treatment, how many people see no improvement and remain unchanged, and how many people see worse outcomes. There are bunch of side questions including length of time and degrees and by the procedures themselves (on which tech is advancing). Then there's also the balance that as of now there are no good mental treatments as well.
As for suicide in trans animals, animals generally do not have a suicide issue. The reason why is because to comprehend death you need to make a few leaps of understanding: 1) you have to understand there is such a thing and what it entails, 2) you have to understand it is a permanent condition, 3) you have to understand its inevitability, 4) you have to understand it applies to you and 5) you have to draw a logical conclusion that it might end whatever suffering you have. There are only a handful of animals that are capable of that (e.g., not dogs). The great apes, the cetaceans, and it looks like maybe a few others like the elephantidae. All of them extremely big brained. The are recorded instances of suicides among the great apes including "death by cop" such as a beta male throwing themselves at the alpha, knowing they have no chance, and the alpha tearing them apart. Mentioned in the De Waal book.