Kinda, but you need to set the incentives right. Both the importer and the overseas manufacturer will have the incentive to underreport the emissions and overreport the paid carbon tax (China could easily collect it on paper, and then offer some rebates).

So you would need to bring together the board of local manufacturers interested in making imports prohibitively expensive, and a board of importers interested in making local production economically unviable. And you would need to have a process where they would need to come out with a compromise satisfying both parties. Both parties should have respect for each other and be willing to find a middle ground. That's how democracies work, that's why U.S. managed to become the #1 economy.

Except, this mechanism is broken. Whoever owns the media can point the finger at the prospective manufacturer, scream "climate change denier" and the public won't listen. Because the public sentiment is that some opinions, depending on the identity/affiliation of their sources, are just too dangerous to be heard. And this is heavily abused by those who know how to play the system.

The public gets screwed long-term. It's already prohibitively expensive to raise kids in the U.S. (good luck competing against 5-people-per-room countries), it's unviable for most people to have retirement savings, or even own their place of residence. But the public doesn't connect it with the fact that the remaining local jobs have zero economic leverage and are made to easily replace people on a whim. We keep screaming at the next "enemy of the people" and stubbornly refuse to notice how wealth and economic leverage is being siphoned away from us.