1) It's extremely unlikely these missiles (or anything) could sink a Nimitz-class carrier. Disable it enough so it can't do flight ops and it's a worthless hunk of metal, however.
2) You can't annihilate China without nuclear weapons, and the nuclear chain of command is securely in the hands of the President. "Field Commanders" can't nuke anybody.
3) I can't conceive of any circumstance where China disabling a carrier with conventional weapons draws a nuclear response.
“extremely unlikely these missiles (or anything) could sink a Nimitz-class carrier”
Man o man, I wish you hadn’t said that.
But, we owe them too much money and their economy depends on us buying their crap. So, I assess a kinetic option as “minimal.”