Arguably, any system of assessing and forcing people to pay for the assessed external costs is at odds with capitalism (it is arguably a form of central planning and interference with the free market).

Which is not to say that it isn't a good idea, and the minimum deviation necessary from capitalism to address externalities, but ideological capitalism/free-market purists tend to oppose it pretty strongly.

"Capitalism" does not mean "free market". It literally means that the profits go to investors (capital) instead of to the workers (labor). You can certainly have a capitalist system that does not have a free market, and you can certainly have a free market system that is not capitalism.