I am not a believer in the 'if you don't do anything wrong you have nothing to worry about' line of thinking, as things can go horribly wrong for people who wouldn't expect it. Nonetheless, the vast majority of transpeople aren't going to find themselves in jail, and nobody with a heart would want those who do to be treated more harshly than other prisoners by being in the wrong jail.
The problem, of course, is with people (AFAIK exclusively men) who claim trans status on arrest, in order to get into women's jails, where they think they might get an easier time, or worse, where they hope to find victims - many of them already have convictions for sexual assault or rape. Chewbacca posted photos of a number of such people on the last thread on this topic that reached 1000 posts and closed, and anyone who didn't read that thread could find them easily - they are towards the end of the thread.
When it comes to the statistics, man identifying as women means that crime figures are skewed, yes. As things stand, rape is a crime that can only be carried out with a penis, but Trans Rights Activists (TRAs) are pushing for it to be redefined as a sexual assault carried out with an object. Both are clearly traumatic for the victim, but to call all of them 'rape' means that a male-only category is subsumed into a gender-free one, so the figures are less nuanced. Similarly, if crimes by transwomen are recorded as female, then all stats for all categories of crime will be corrupted.
The fact that the men in Mollygo's link have reverted to male after release from jail can only suggest that their trans status claims were deliberately staged - something that has concerned many women for some time.