They don't have to consent. That's the whole point of tariffs. They can be imposed unilaterally (As we have recently seen with the US - China shoving contest. Which, of course, had nothing to do with the environment, or carbon emissions, and everything to do with political optics.)
The purpose of carbon tarrifs is not to punish other countries.
The purpose of carbon tarrifs is to make foreign dirty-energy-intensive products less competitive, compared to less dirty-energy-intensive ones.
Because we already have a way of making domestic dirty-energy-intensive products less competitive. It works great. It's called a domestic carbon tax.
They remove the immediate complaint of "If we impose a local carbon tax on energy, foreign imports will outcompete us, because they don't have to pay it."
A tariff is generally across the board though, by product type. If one company starts being more "green" than the others in the country of origin, it doesn't generally get hit less by tariffs.
This is Tom Clancy fairy-tale nonsense. China wants a nuclear-free Korea, for the same reason that the United States wants a nuclear-free Mexico and Canada. Nuclear super-powers aren't keen on letting their neighboring satellite states limit their foreign policy, by arming them.
A carbon tax is not 'stopping trade'. It's an increase to the price of goods that are energy-intensive to manufacture, or ship.