do you accept that some people firmly believe they are born in the 'wrong' body?
Body dysmorphia is a thing. Some people with entirely unremarkable noses firmly believe they're huge and ugly. Some people starving themselves to death are sincerely convinced they're disgustingly fat. Some people believe they have the wrong number of legs and are deeply distressed by the extra one.
But believing something, no matter how firmly or sincerely, doesn't make it true. You are born in - and as - your body. You can't accidentally end up in someone else's, because then you would be them instead of you.
and have a right to live as the other sex?
You can't. It's not a matter of rights, it's a matter of reality. You cannot change sex, therefore you cannot live as the opposite sex.
How about gender? Well ... that's where it gets complicated. Because gender shifts, and is made up of 2 artifical boxes that nobody really fits into.
My grandfather smoked a pipe and fought with the Gurkhas. He also embroidered beautiful floral tablecloths and taught my mother to knit.
My grandmother was also keen on knitting and embroidery, she also joined the army. In common with most of her regiment, she rejected the uniform dress in favour of trousers, because when you're clambering in and out of trucks on the front line of the liberation of France a dress is impractical and the male soldiers can see your knickers. In later life she reverted to skirts and taught herself quantum physics for fun.
My other grandmother was never in the army, was obsessed with housework, and I never once saw her in a skirt or dress - only trousers.
Both grandmothers had short hair and wore flat shoes.
What gender were any of them living as?
Should it be as unremarkable for a man to have long hair and wear makeup and a skirt as it is for a woman to have short hair, no makeup and trousers? Yes. It was good enough for the ancient Greeks and Egyptians, so why not now? Should it be OK for women to enjoy brewing beer? Yes - beer used to be considered a women's thing, not a men's one, in medieval Britain. Should people be allowed to do any job, have any hobby or interest regardless of their sex? Of course.
Is it OK for men to cry and talk about their feelings? Or for women to express their anger? Yes - much healthier all round.
But there's no ned or reason to pretend to be the opposite sex in order to do any of these things.
GC feminism is about breaking down those boxes so people can live how they choose, regardless of sex. Gender ideology is about keeping and reinforcing the boxes, but allowing a few people to jump between them.
possibly taking hormones and undergoing surgery?
Hormones and surgery are extremely serious medical interventions, with substantial adverse effects even if they go 'well'. When they go badly - which is often - the consequences are horrific.
So they absolutely should not be available to children. If they are in psychological distress they should get rapid and thorough psychological intervention to understand the source of and attempt to alleviate that distress. For the vast majority, this exploratory therapy (and for lower levels of distress, simply supportive watchful waiting) has been shown to be sufficient for it to resolve.
Adults? More difficult. We let people with body dysmorphia about their nose have cosmetic surgery (although rarely on the NHS). We don't give gastric bands to anorexics, or let people chop off a healthy leg - even privately.
So I think for adults - after extensive psychological assessment and therapy - there are some circumstances in which allowing it might be reasonable, and many in which it would remain unethical. Exactly where I would draw the line is tricky.
Certainly if done it should be with much better information and long-term follow up than is currently the case. And if long-term studies show an intervention is not effective (the small amount of data available suggest surgery doesn’t improve, and may worsen, long term psychological outcomes) then it should be reassessed.
Or [that] these people should not ever be supported to live as the opposite sex? That doing so is always wrong?
As above it's not so much a question of 'wrong' as impossible/illogical.
People should be allowed to live as they choose - provided there isn't an adverse effect on others. But there's no reason to give additional support to someone 'living as' the opposite sex (or gender) - whatever they perceive that to mean - than there is to support any other choices they make about their life.