They are not externalising their cost to the rest of society. They are paying the exact cost of the energy they spend just like the rest of people.
[0] https://www.umweltbundesamt.de/daten/umwelt-wirtschaft/gesel...
[1] https://link.springer.com/article/10.1007/s10584-020-02785-4
Also I don't see why only a small part of society should be paying for that "carbon tax" - all citizens should pay an amount corresponding to their energy spending and then we are back to miners not externalising anything.
Where did I say that only a small part should be paying the real cost? That's a straw man.
And no: miners are externalizing exactly because the lack of a carbon tax. And from my experience they are not very interested in introducing one either.