Instead, it should be paired with a dividend which makes it revenue neutral for the median household. Lower income families, who are more impacted by but less responsible for climate change, would be paid a benefit.
I'm not sure how these two ideas make sense in your head. If you implement a carbon tax and the rich are the greatest contributors as you stated, one would expect them to be hit by the carbon tax as well.
In any case, I personally see an income tax as a strange thing to have in the context of a discussion wherein taxing undesirable behavior is seen as a way of disincentivizing it.
At the very minimum free the working class and lower-middle class from income tax.