I understand what you're saying.
Can you separate hatred of someone's actions from hatred of the person? That's what you're seeing here.
Very, very few Mumsnetters - or people in the UK, I would guess; we're still a tolerant nation - actually hate trans people.
Those who hate the existence of trans people tend to be vehemently conservative individuals, willing to fight transgressions of their rigid beliefs in sex-limited social roles. Most such people are religious fundamentalists, more likely to be found in America or the Islamic world than here.
Britain, like the rest of Western Europe, suffered hundreds of years of religious intolerance and is deeply committed to steering clear of that shit. We ditched our residual homophobia fifty years ago now: you'd be pushed to find anyone so defensive of masculinity that they revile gay or feminine-leaning men. People like that still exist, of course, but their views are unpopular.
The transphobia of which you speak isn't hatred of men who feel like they're women. We don't hate women who feel like royalty, children who feel they are horses, or the messed-up guys on the Tube who feel they're being followed by demons. Our collective response is a shrug. If we're in a good mood, we might ask them if they're okay. That's about it.
We expect equivalent tolerance. If the 'royal' woman demands obeisance, we refuse. She's free to believe we are her subjects, but not to force others to role-play with her. If someone played along with the child by saddling her up, we'd report them for cruelty. The haunted guy may believe his fellow passengers are demons, he may not attempt to exorcise them. The trans woman's free to believe s/he has changed sex, not to require everyone else to agree they really have.
I don't believe it is phobic or hateful to say "Yes, I understand this is how you feel. If it makes you happy, great! I'll ask you, though, to understand that my sex is genetically female and some things are limited, for reasons of safety and fairness, to people of one genetic sex or the other." They may find it a bit upsetting but, in my experience, generally accept that this is reasonable.
The ones who call it transphobic are demanding that everyone must role-play along with them. This is unacceptable: they are, effectively, requiring me to deny my own sex, my physical reality. In doing this, I'd have to surrender my rights to privacy, fairness and things like medical safety (and more, because being physically female has ramifications in all areas of life).
I've never hated another person in my life. I think it is reasonable to hate beliefs and behaviours that harm me (or other people) or seek to impair my rights. I've met a couple of influential Muslim leaders. I'd say they are good men: intelligent, thoughtful, morally observant. They believe in FGM. I respected who they are as individuals, enjoyed their discourse on relative rights and obligations, and I fucking hate their commitment to cutting out women's clitorises. You can understand that, can't you?
We can respect the human being, accept that their beliefs are sincere, and hate the effects of their beliefs.