1) The numbers don't work out. Even a very steep CO2 tax will produce too little revenue for any form of meaningful BI.The USA currently consumes about 450 billion gallons of petroleum products per year, which is about 1.5 trillion pounds. We also mine about a billion short tons of coal per year, which is 2 trillion pounds. We have about 300 million people.
Taxing $0.75/pound on carbon fuels would generate a bit over $6k/year per man, woman and child. That's $4.5/gallon. This is enough to put a family of 4 at about $25K/year in basic income, which is just above the poverty line this year.
There is enough revenue. You just have to accept a higher tax.
2) The money collected on the CO2 tax could also be used for a general tax cut. So you still have the political problem of explaining to working middle class why their money is being handed out to lazy bums.
The middle class receives the money as well. This kind of program - where everyone pays and everyone receives - has proven very popular in the past. See social security for a demonstration.