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Oh, don't give me that shit. The point is that genetics alone is not sufficient to determine sex. Genes, the expression of genes, exposure to hormones in the uterine environment, the body's sensitivity to sex hormones all play a factor in sex, gender, and sexual orientation. There are a lot of steps along the way where sexual development can be influenced and trying to reduce everything to chromosomes is reductionist.

I can only speak for myself, but my willingness, or lack thereof, to conform to gender roles remains separate and distinct from my identity as a male. I never once considered that I'm not a male because I liked some things that were "feminine" or disinterested in some things that were "masculine". I trust transgender individuals can understand the difference between gender roles and gender identity as well.

"immutable" as in "nothing can change it"?

You ought to look into CRISPR.  I can say without a hint of hyperbole that it's one of the most powerful technologies humanity has stumbled on because it can be used to decided after the fact what any part of a genome can be.

If you aren't into dry research papers, here's a fun episode of John Oliver's "Last Week Tonight" that took a peek: Link

Can CRISPR change a human's sex after they've been born? Or is it something that's done - or can be done, if the technology is still being developed - to the embryo while it's in the womb?

If it does become possible to physically change sex, I wonder how many transpeople will use it? Something that annoys, or even infuriates, many feminists about transwomen is that they seem to retain their male biology, and reduce being female to pretty superficial characteristics - wearing make-up and a wig, and putting on a dress.

At the moment, no.  Experiments on humans for procedures outside curing disease may be frowned on for some time too (maybe even in China).  Obviously any edits would be vastly simpler on a single cell like a fertilized ovum. However, the technology is still being aggressively developed. Due to it's utility in research, the technology has improved several generations in just a few years, an amazing pace of progress.  

That said, there's things like this: Link.  and more recent update (Link).  The inevitable, "yeah, but that's mice" caveats apply, but perhaps far less brute-force tinkering is required than one might imagine.  The main point being that thanks to CRISPR, it may not be too long before even genetic level biological attributes become far more mutable than had ever been possible because we simply didn't have a tool.