Sending the money to every man woman and child can't be a long term goal for improving GHG emissions so I'm assuming there is a game the government is playing here.
You aren't echoing me, I don't think. Sending the money back to everyone is exactly the ideal implementation of sin taxes. The idea is to make the bad behavior relatively more expensive without affecting the average person's net income.
So, for a carbon tax, the way it works is everyone pays an extra dollar per liter of fuel. Then, the government sums up everything that was paid, and sends the average amount paid back to each household. After receiving the rebate, a Prius or Leaf owner should actually have
more money than they started with, because they burned so little carbon relative to the average. But the average F150 owner will have less money.
The idea is to incent that F150 owner to consider how painful visiting the pump is, and maybe choose a different car next time. Or to encourage the hybrid and electric car owners in their decisions ("Oh, man, this gas station visit would have been much more expensive if I had a bigger car! Sure am glad/will next time buy a similar fuel sipping vehicle."). You can do all this without affecting the average consumer's wallet at all, but at the same time, you've created an incentive for every consumer to reduce their carbon impact.
In theory, this should be sufficient to drive any amount of carbon reduction desired: you just gradually ratchet up that tax until the total carbon emission rates are what you need to accomplish your policy goals. But we won't do that, because humans suck at long term thinking.