After a nuclear exchange with China, there would be no "USA" for you to fight for. Just like there is no "Atlantis" or "Cherokee Nation" today. China has enough fissile material for at least 1000 plutonium (strategic, not tactical) nukes. At least that figure has been produced by the two known nuclear facilities in China. In such an exchange, all the urban population of the USA would be vaporized. Welcome to the stone age.
Good to see you again Roger.
It certainly is very likely that China would unleash nuclear weapons on the United States in that situation.
However you have to understand that in that situation, China will be more throughly destroyed then even the US. Obviously China's leaders would not want that to happen. It is much more likely that they would use nuclear weapons against a beachead and/or invasion fleet. Or they could not use nuclear weapons at all, rather allowing a US expeditionary force to be destroyed by vastly superior numbers of Chinese troops and China's size.