That's a fundamental problem with any regulatory measure intended to reduce carbon emissions. Cheap energy is good for the economy. Fossil fuels are cheap. Any attempt to reduce the use of fossil fuels is going to hurt voters in the short term no matter how you package it.

Sure, you can try to be sneaky about it, implementing policies that only indirectly impact fossil fuels and hoping the voters don't notice you're the one to blame for the rising price of energy. Or you can be honest, bite the bullet, and try to convince the public a carbon tax is necessary to avoid something even worse down the line. Either way you're fighting an uphill battle.