If a boy was raised as a girl since very young, complete with female pronouns, socialisation, given a new female body, would he be happy living as a woman?

It doesn't work.

There's a group of boys born without penises, look exactly like female babies externally and who then get raised as girls. The penises grow much later.

Most of them get spotted as boys long before puberty because if their behaviour, and a lot will outright state they are boys and want to be dressed and present as boys. Virtually all have a male gender identity as adults. It's a rare condition, out of about eighteen cases in one study and seventeen had a male ID.

And yes you can tell if a brain is male or female. About the only time it's uncertain is if the person is gay.

There are also boys with severely deformed genitals who get surgically altered to look like girls, got raised as girls. A look at them as adults showed that almost all reverted to male ID.

So it does seem it's innate.