Even then, the principle still stands. If you want to pay off government debt, budget for it and pay it out of general revenues (minus carbon tax) or cut spending. Don't tie to a specific tax from a specific policy - because then you're incentivizing government behaviour that is not conducive to what the tax is trying to solve, like the fact that you may want to raise or lower the carbon tax not because it is good environmental policy, but because it is good for lowering the deficit (or funding more rooftop gardens or whatever) - which defeats the purpose!
Worse, once you start relying on this revenue, and this revenue goes down (as people use less carbon) - you're going to get yourself in trouble.
This reminded me of the counties (or whatever) that became reliant on fines from big tobacco. The incentive for the tax collector is, absurdly, to keep the negative externality going.
Maybe good economic policy should examine both ends of the wealth transfer: is it good that we take money from eg. big oil, and meanwhile also good that we put the money in the govt. budget? But most political talking points are one sided: “X should pay” or “Y should get paid.”
Worse, once you start relying on this revenue, and this revenue goes down (as people use less carbon) - you're going to get yourself in trouble.