The company being tariffed suffers as well, so long as the tariffs are enforced by a large number of countries, and they don't have a lock on the market (like specific rare elements or something).

"Your own" citizens bear the burden of higher prices from that country, but this is a desired outcome: goods that have higher externalities should cost more. That's the whole point of a carbon tax.

And unless a country really has a lock on a resource, higher tariffs ought to mean the products start being produced in the host country, which is good because it will be being made under the better carbon policies.

Tariffs are dumb when they're enacted out of spite, enacted out of some false sense of balance, or enacted as a short-term bandaid to try and prop up a failing local industry. But in the case of carbon taxes and policies, they would seem to do exactly the right thing: discourage consumption and production of carbon-heavy resources.