But if the invasion has already begun....
How many American lives would we sacrifice for Taiwan?
Such an invasion could not be made in a vacuum, and would be known long before it happened. And the moment such things were obvious, that is when you start your own movement of pieces.
To begin with, their Navy is in no way set up to do this, and neither is their Army. You would have months of preparation first, China could not hide that. And weeks before such could happen, they would have to start bringing all of their pieces together on the board. Loading ships, and moving them to positions to prepare for the attack. Also the forces needed, those also would have to be moved into position.
This would be seen, and the US would be acting in concert at the same time to let China know it was watching.
And if I had to predict, if they tried it, largely it would end up being a three ring circus.
And the reason is simple, China's Navy is almost a joke. They do almost no fleet operations, and have no experience in amphibious operations. I bet most of what they do "know" is out of the books on that the US wrote 70 years ago. There is a hell of a lot more involved than simply "put people on boats, sail boats, put soldiers on ground". That was not even how it was done 100 years ago, as the UK forces discovered at a little place called Gallipoli.
No, the US would be watching each of the steps they started to make, and would be countering them with their own forces. Most likely simply by placing them into positions so they would have to "go through" the US in order to get to Taiwan. And we all know how that ends up, we saw it in 1941.
When Japan wanted the resources of the Dutch East Indies, but realized that to get through them, they had to go past a stronghold of the US. They made the decision to attack, and the world knows the result. And just like in 1941, the UK was heavily involved because they see that as "their area of the world" as well because of alliances and partnerships in the region. As well as France.
Myself, I often shake my head when people for some reason think it would be the "US alone", because many nations would be involved. Including Japan, Philippines, South Korea, Australia, possibly even Russia. And not on the side of China, Russia would also see this as a threat to their interests in the area. And at each step China started to make, others would see it and be making their own movements.
As far as "How many American lives?" Well, ultimately that is up to China. But for myself, I will say it this way and do not take it as an attack. I am not such an arrogant and self-centered prick that I measure lives in that way. If a nation makes a naked show of force and tries to annex and destroy another country then I have no problem in fighting against it. Even at the risk of my own life, which I proved by my own actions in the past.
Because if there is one thing we should have learned in the last century, appeasement does not work. Allow a bully to do it once, they will simply keep doing it over and over again. And they must be stopped quickly, or it will only end up worse in the long run.