Except, how do you get to choose what type of power you buy? What if I live back east and would happily pay for wind or solar but my only option is coal or natural gas?
In many states back east (at leas Massachusetts and New Hampshire), you can choose who is your energy supplier, as the distributor and provider are separate items on the bill. There are several "green energy" suppliers vying for your business.
This is great but has the unfortunate side-effect of some energy suppliers telemarketing to try to get you to switch to their "lower rates", which inevitably increase again after a few months.
So check the system in your state, you may be pleasantly surprised.
Remitting on a per capita basis doesn't change the regressive nature. You can only fix the regressive nature if you return it based on income. And even then, the poorest people don't have the float necessary to pay for the bill now and get a tax refund later.
Wealthy people typically have a larger carbon footprint than those who aren't well off. Think flights, buying a car more frequently, etc.
If everyone has the same carbon footprint, the tax is strictly neutral. If people have different footprints, money flows from polluters to non-polluters.